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Slide1
Year 7 Number Theory
Dr J Frost (jfrost@tiffin.kingston.sch.uk)www.drfrostmaths.com
Last modified:
26
th November 2015
Objectives:
Have an appreciation of properties of integers (whole numbers), including finding the Lowest Common Multiple, Highest Common Factor, and using the prime factorisation of numbers for a variety of purposes
. Reason about divisibility in equations.Slide2
For Teacher Use:
Recommended lesson structure:
Lesson 1
: Introduction to Number Theory/Sums of
primes+squares
problems.
Lesson 2
: Prime Factorisation
Lesson 3
: LCM/HCFLesson 4: Uses of Prime FactorisationsLesson 5+6: Divisibility RulesExtension: Divisibility of Terms/Within Equations
Go >
Go >
Go >
Go >
Go >
Go >Slide3
Starter
List the following numbers in your books.
Bro Pro Tip
: You should try to memorise these.
The first 16
square numbers
:
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256
The first 8
cube numbers
:
1, 8, 27, 64, 125, 216, 343, 512
The
prime numbers
up to 40:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37The first 10 triangular numbers
:(e.g. 3 is a triangular number as you can form a triangle using 1 dot on the first row and 2 on the next)1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66
If you finish:
A ‘
perfect’ number
is a number who factors (excluding itself) add up to itself.
For example. The factors of 6 (excluding 6) are 1, 2, 3, and
.
Find the first perfect number after 6.
Solution:
?
Bro
F
act
: All perfect numbers are triangular numbers.
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Other numbers you might want to investigate yourself:
Tetrahedric
numbers,
Fibonacci numbers.Slide4
Key Terms
!
Integer
: A whole number.
Positive integer
: An integer that is at least 1.
Non-negative integer
: An integer that is at least 0.
Perfect square
: For integers, just a square number.
Divisor: Another word for factor.Composite: The opposite of prime: has other factors.
Bro Side Note: A ‘perfect square’ more generally refers to ‘something squared’, which can be an algebraic expression.
For example
is a ‘perfect square’, but is not necessarily a square number, e.g. if
(as
)
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!
Distinct integers:
Numbers which are different!
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Puzzles involving sums of primes/squares/…
Example:
Goldbach’s
Conjecture (as of current, unproven!) states that all even numbers greater than 2 are the sum of two primes.
How many ways are there of expressing 100 as the sum of two primes?
Solution: 6 (3 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, 47 + 53)
Bro Tip
: It often helps to write out your numbers of interest (primes, squares, …) first.
?
Further Example:
The Indian mathematician Ramanujan once famously noted that the 1729 number of a taxi ridden by his friend Hardy:
“
is a very interesting number; it is the smallest integer expressible as a sum of two different cubes in two different ways”.What is the smallest integer (not necessarily a square) that is expressible as the sum of two distinct squares in two different ways?
(Hint: 1 is used in one of the sums)
(Side note: the smallest
square number
expressible as the sum of two squares in two different ways is
)
?Slide6
[JMC 2015 Q11] What is the smallest prime number that is the sum of three different prime numbers?
A 11 B 15 C 17 D 19 E 23
Solution: D[JMO 1999 A2] In how many different ways can 50 be written as the sum of two prime numbers? (Note:
and
do not count as different.)Solution: 4 ways (
)
[JMO 2009 A3] The positive whole numbers
and
are all different and
. What is the value of
?
Solution:
17
1
2
3
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Exercise 1
(Problems on provided sheet)
Bro Tip
: Again, use your lists of numbers from the starter.Slide7
[
JMC 2015 Q19] One of the following cubes is the smallest cube that can be written as the sum of three positive cubes. Which is it?A 27 B 64 C 15 D
216 E 512Solution: D
[JMC 2006 Q20] The sum of three different prime numbers is 40. What is the difference between the two biggest of these numbers?A 8 B 12 C 16 D 20 E 24Solution:
E[JMC 2010 Q22] Kiran writes down six different prime numbers,
, all less than 20, such that
. What is the value of
?
A 16 B 18 C 20
D
22 E 24 Solution: E
[TMC Regional 2009 Q9] 12345 can be expressed as the sum of two primes in exactly one way. What is the larger of the two primes?Solution: 12343. Note that odd = odd + even only. Thus one of the two primes must be 2.
Exercise 1
4
5
6
7
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Important Note
: If three numbers sum to an even number, they can’t all be odd. But 2 is the only even number, so must be one of the numbers. Slide8
[JMO 2006 A9] The prime number 11 may be written as the sum of three prime numbers in two different ways:
and
. What is the smallest prime number which can be written two different ways as the sum of the three prime numbers which are all different?Solution: 23
[JMO 2014 B6] The sum of four different prime numbers is a prime number. The sum of some pair of the numbers is a prime number, as is the sum of some triple of the numbers. What is the smallest possible sum of the four prime numbers?
Exercise 1
N
N
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?Slide9
Prime Factorisation
To find the prime factorisation of a number is to express it as a
product
of
prime numbers
.
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Bro Tip
: While
is also correct, we can use ‘index notation’ to group prime factors together that are the same.
120
20
6
4
2
2
5
2
3
We can use a ‘tree’ to help us with the working.
For each number, find two numbers the multiply to give it.
?
If you get to a prime, we can’t branch out further, so we have a ‘
leaf
’. It’s helpful to circle the leaves.Slide10
Another quick example
2250
225
10
45
5
5
2
5
9
3
3
Possible Tree
?
?Slide11
Check Your Understanding
Using a tree, find the prime factorisation of 1350.
When done, try coming up with more trees. What do you notice about the final result in each case?
1350
10
135
2
5
5
27
3
3
1350
270
2
5
5
90
3
3
30
3
9
6
3
We always end up with the same leaves each time, and hence the same factorisation.
Fundamental Law of Arithmetic/Unique Factorisation Theorem
: Every positive integer can be
uniquely
expressed as a product of primes.
Some Possible Trees
?
?Slide12
Prime Factorising a number already in index form
Sometimes you might have a number with powers, but the base (the big number) is not prime. How would you prime factorise this? What if a base was repeated?
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Quickfire
Questions:
N
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Bro Note: This is an example of a ‘law of indices’, which you will learn more about in Year 8.
Working:
Slide13
Exercise 2
By drawing a tree of otherwise, find prime factorisations (in index form) for the following numbers.
Is
odd or even?
An odd number to any power is always odd.
Put in prime factorised form:
Prime factorise the following:
Suppose 1 was considered to be a prime number. Explain why this violates the Fundamental Law of Arithmetic.
For example, 6 could be expressed as
or
or
. But FLA states there is a
unique
factorisation for each integer. Thus 1 is not prime.
[TMC Regional 2012 Q4] Find the sum of all numbers less than 120 which are the product of exactly three different prime factors
.
Solution: 717
1
2
3
5
N
1
N
2
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4
What are the factors of
? Give your answers in index form.
?Slide14
Starter
List the factors of
(keeping your factors in prime factorised form)
One number will be a factor of another if the prime factor(s) are the same but the powers are smaller (or equal).
List a few multiples of
which only contains prime factors of 3.
(keeping your multiples in prime factorised form)
divides for example by
because
You will learn in Year 8 that
, so this is consistent with the pattern.
One number will be a multiple of another if the prime factor(s) are the same but the powers are greater (or equal).
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?Slide15
Lowest Common Multiple/Highest Common Factor
Multiples of 8: 8, 16, 24, 32, …
Multiples of 12: 12, 24, 36, …
Lowest Common Multiple of 8 and 12: 24
Factors of 8: 1, 2, 4, 8
Factors of 12: 1, 2, 3, 4, 6, 12
Highest Common Factor of 8 and 12: 4
For small numbers, we can list out multiples of the larger number until we see a multiple of the smaller number.
For small numbers, we can list out factors of each number and choose the greatest number which is common.
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Check Your Understanding
Bro Shortcut
: Any multiple of 60 ends with a 0. Therefore the multiple of 72 must be x5, x10, …
Bro Shortcut
: Any number which goes into 60 and 72 must also go into their difference! (i.e. 12)
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?Slide17
But what about bigger numbers?
792, 378
Sometimes it’s not practical to use this method.
Can we use the prime factorisation somehow?
?
?Slide18
But what about bigger numbers?
Alternative
: Venn Diagram Method
3
3
2
2
2
7
11
Method
: ‘What wins what loses’
792
378
Step 1:
Find the prime factors common to both numbers.
Step 2:
Fill in the remaining prime factors of each number.
HCF is product of numbers in the intersection.
3
LCM is all numbers multiplied.
?
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Step 1:
Align numbers so that each prime factor has its own column.
For HCF, out of
and 2, what factor is common to both? We saw from earlier that 2 is a factor of 2 and
.
We effectively find what ‘loses’ out of 2 and
. In this case 2.
We see what ‘loses’ in each column (where ‘nothing’ always loses against ‘something’)
For LCM, what is both a multiple of
and
? Again, from earlier, they both go into
,
i.e. the one that ‘wins’
. Repeating for the other numbers:
Slide19
More Examples
(note that if there’s a ‘draw’, both win and both lose)
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Line numbers up:
?Slide20
Check Your Understanding
If you finish…
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?Slide21
Exercise 3
Find the LCM and HCF of the following pairs of numbers (using any suitable method).
6 and 8
HCF = 2, LCM = 24
13 and 5
HCF = 1, LCM = 65
12 and 15
HCF = 3, LCM = 60
21 and 35
HCF = 7, LCM = 105The K4 bus comes every 9 minutes. The K3 bus comes every 12 minutes. If they both come at 9am, at what time will they next arrive at the same time?9:36amFind the LCM and HCF of the following pairs, by prime factorising the numbers first.
a) 36 and 378 LCM = 756, HCF = 18b) 315 and 3675 LCM = 11025, HCF = 105c) 72 and 66 LCM = 792, HCF = 6
d) 2880 and 792 LCM = 31680, HCF = 72e) 375 and 325 LCM = 4875, HCF = 25f) 252 and 2079 LCM = 8316, HCF = 63
[JMC 2009 Q18] Six friends are having dinner together in their local restaurant. The first eats there every day, the second eats there every other day, the third eats there every third day, the fourth eats there every fourth day, the fifth every fifth day and the sixth eats there every sixth day. They agree to have a party the next time they all eat together there. In how many days’ time is the party?
60 days[IMC 2013 Q15] I have a bag of coins. In it, one third of the coins are gold, one fifth of them are silver, two sevenths are bronze and the rest are copper. My bag can hold a maximum of 200 coins. How many coins are in my bag?
A 101 B 105 C 153 D 195Solution:
B (LCM of 3, 5, 7)
finds the number of integers between 1 and that share no factors with
other than 1 (i.e. the HCF is 1).For example
because for two numbers up to 6,
1 and 5, HCF(6,1) = 1 and HCF(6, 5) = 1.What is
?
4
What is
?
6
What in general is
for a prime number
?
Given that
(provided that
), find
Bro Note:
’s posh name is ‘Euler’s Totient Function
’.
Calculator permitted!
1
2
3
4
5
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Year 7 Uses of Prime FactorisationsSlide23
Other uses of prime factorisation
We have so far seen that prime factorisations are helpful to find the Lowest Common Multiple and Highest Common Factor of two numbers.
Use #1
: Square/cube numbers
Example: “What is the smallest multiple of 504 that is square?”
Use #2
: Number of factors
Example: “How many factors does
have?”
Use #3
: Trailing 0s (extension)
“How many zeroes are at the end of
?”
“What is the last non-zero digit of 20! ?”
Slide24
Use #1
: Square/Cube Numbers
Find the prime factorisation of the following
square
numbers. What do you notice?
Find the prime factorisation of the following
cube
numbers. What do you notice?
!
Square numbers have even powers in their prime factorisation.
!
Cube numbers have powers which are multiples of 3.
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Use #1
: Square/Cube Numbers
What is the smallest multiple of 504 that is square?
(Note that
)
The power of 3 on the 2 and the power of 1 on the 7 are both odd.
So we need to multiply by 2 and 7, i.e. 14, to give 7056.
Q
?
Check your understanding:
What do you need to multiply the following by to make the following square and cube?
Square
Cube
Square
Cube
How many square numbers are factors of
?
All
even powers of 2
(
).
But don’t forget that
(
)
is square
! (as the 0 is even). That’s
6 square factors
.
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a
b
Bro Reminder
: Factors of
will be all powers of 2 with a power up to 10.
Slide26
Use #2
: Numbers of factors
Note that
. Write all the factors of 72, each with their prime factorisation.
(no prime
factorisation!)
How many possibilities were there for the power of 2 in each factor?
4 possibilities (powers of 1, 2, 3 or doesn’t appear)
How many possibilities were there for the power of 3 in each factor?
3 possibilities (powers of 1, 2 or doesn’t appear)
Therefore how many possibilities are there overall?
There are therefore 12 factors.
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!
To find number of factors, +1 to each power and multiply these powers.
Bro Reminder
: We’ll have a factor of 72 provided the powers of 2 and 3 are less than or equal to the powers in
Slide27
Use #2
: Numbers of factors
How many factors do each of these square numbers have?
Number:
4
9
16
25
36
Prime
Fac
:
Num
Factors:
3
3
5
3
9
Number:
4
9
16
25
36
Prime
Fac
:
Num
Factors:
3
3
5
3
9
What do the number of factors all have in common?Square numbers have an odd number of factors.
Why is this?Square numbers have even powers. To find the number of factors, we add one to each power and multiply. Adding 1 makes all the powers odd. And multiplying odd numbers together gives an odd number.
!
Square numbers (and only square numbers) have an odd number of factors.
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Examples
Find how many factors
has.
Num
factors
How many factors does
have?
Num
factors
[
JMC 2000 Q23]
A
certain number has exactly eight factors including 1 and itself. Two of its factors are 21 and 35. What is the number?
A 105 B
210
C 420 D
525
E 735
Note that
and
.
So the number must be some multiple of
. But this does have 8 factors, as
.
It’s helpful to put 1 as the power even though we usually wouldn’t.
Q
Q
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Check Your Understanding
How many factors does 90 have?
Num
factors
How many factors does
have?
Num
factors
A number has 3 factors. What type of number is it?
If it has 3 factors it must be of the form
where
is prime. It is therefore a square number (e.g. 25), and more specifically, the square of a prime.
a
b
N
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Exercise 4
(on provided sheet)
1
2
3
4
6
5
7
8
9
10
N
1
Use the following prime factorisations (if provided) to find the smallest number we need to multiply the number by to make it (a) a square (b) a cube.
(
i
)
Solution
: (a) 6 (b)
180
(ii)
Solution: (a) 6 (b) 9
(ii) 16200
Sln
:
(a)
2
(b)
45
List the cube
numbers (in prime factorised form) that are factors of
. Solution: 1,
Use the fact that
to list out all the factors of 60 (excluding 1) in prime factorised form.
By using the
prime
factorisations, determine how many factors each of the following numbers have.
Solution
: 9
Solution
: 6
Solution: 12
Solution: 16
[
JMC 2012 Q3] Which of the following has exactly one factor other than 1 and itself?
A 6 B 8 C 13 D
19 E
25
Solution:
E
[IMC 1999 Q16] Three statements:
i)
is even (ii)
is odd (iii)
is square
Exactly which ones are true?
A (i) only B (ii) only C (iii) only
D (i) and (iii) only E (ii) and (iii)
Solution:
E
[JMO 2008 A6] How many positive square numbers are factors of 1600?
Solution
:
8
How many factors do the following numbers have?
Solution
: 64
Solution
:
121
How many numbers between 1 and 16 have an odd number of factors?
Solution: 4 (as four squares
)
[Junior Kangaroo 2015 Q23] How many three-digit numbers have an odd number of factors?
A 5 B 10 C 20 D 21 E 22
E (22 squares between 100 and 961
)
[JMO 1997 B2] Every prime number has two factors. How many integers between 1 and 200 have exactly four factors?
Solution
:
59. To have four factors the number has to be of the form
, or the product of two different primes.
[Cayley 2013 Q1] What is the smallest non-zero multiple of 2, 4, 7 and 8 which is a square?
Solution:
784
[JMO 2012 B2] Anastasia thinks of a positive integer, which Barry then doubles. Next, Charlie trebles Barry's number. Finally, Damion multiplies Charlie's number by six. Eve notices that the sum of these four numbers is a perfect square. What is the smallest number that Anastasia could have thought of? (Hint: make Anastasia’s number
)
Solution: Numbers are
which add up to
. Since
, then
could be 5 to give
which is square. Thus Anastasia’s number was 5.
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3Slide31
Supplementary Questions
(for teacher use)
What are the factors of
that are both square and cube? Leave your answer in factorised index form.
Solution: 1,
How many positive factors does
have?
Solution: 325
A certain number has exactly eight factors including 1 and itself. Two of its factors are 33 and 15. What is the number
?
Solution: 165
[SMC 2003 Q15] The number of this year, 2003, is prime. How many square numbers are factors of
?
Solution: 1002
[Kangaroo Pink 2010 Q18] How many integers
, between 1 and 100 inclusive, have the property that
is a square number?
A 99 B 55 C 50 D 10 E 5Solution: B
[
JMO Mentoring Jun2011 Q2] How many positive divisors does 6! have including 6! and 1? [
.]
Solution: 30
[JMO 1996 B1] How many positive whole numbers up to and including 400 can be written in exactly one way as the product of two even numbers
?
Solution: 25 (each product must be of the form
where
is prime)
1
2
3
4
6
5
7
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Year 7 DivisibilitySlide33
2
Last digit is even.
3Digits add up to multiple of 3. e.g
: 1692: 1+6+9+2 = 18 4Last two digits are divisible by 4. e.g. 143328
5Last digit is 0 or 5.6
Number is divisible
by 2 and 3 (so use tests for 2 and 3).
7
Double the last digit and subtract it from the remaining number, and see if the result is divisible by 7. e.g
: 2464 -> 246 – 8 = 238 -> 23 – 16 = 7. 8Last three digits divisible by 8.9Digits add up to multiple of 9.10Last digit 0.11
When you sum odd-positioned digits and subtract even-positioned digits, the result is divisible by 11.e.g. 47949: (4 + 9 + 9) – (7 + 4) = 22 – 11 = 11, which is divisible by 11.12
Number divisible by 3 and by 4.
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Divisibility Rules
How can we tell if a number is divisible by:
!Slide34
Quickfire
Divisibility
4
6
7
9
11
726
168
9196
252
1001
91
216
87912
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Quickfire
Mental Primes
Apart from the obvious instant checks (divisibility by 2, 5), we usually only have to mentally check
3, 7 and 11
to have a good ‘guess’ that a number is prime.
3
7
11
Is it prime?
91
No
101
Yes
234567
No
131
Yes
781
No
751
Yes
221
No! (
)
3
7
11
Is it prime?
91
No
101
Yes
234567
No
131
Yes
781
No
751
Yes
221
N
For 221, what is the largest prime we would have had to test divisibility until we’d be certain it was prime?
Up to
because all composite numbers have a factor (other than 1) up to the square root.
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Advanced
: 13 trick is “Quadruple last digit and add to remaining number. Is result divisible by 13? Slide36
[JMC 2012 Q23] Peter wrote a list of all the numbers that could be produced by changing one digit of the number 200. How many of the numbers on Peter’s list are prime?
A 0 B 1 C 2 D 3 E 4
Solution: A
Test Your Understanding
[JMO 1997 A5] Precisely, one of the numbers 234, 2345, 23456, 234567, 2345678, 23456789 is a prime number. Which one must it be?
Solution: 23456789
Easier One:
Harder One:
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?Slide37
Breaking Down Divisibility Rules
Are these statements true or false?
If we want to show that a number is divisible by 15, we can show it is divisible by 3 and 5.
If we want to show that a number is divisible by
24,
we can show it is divisible by
6
and
4.
False
True
False
True
The problem is that 12 is divisible by 6 and 4, but it is not divisible by 24!
We need to pick two numbers which are
coprime
, i.e. do not share any factors.
How can we therefore test if a number is divisible by 24?
!
Two numbers
and
are coprime if
Slide38
Quickfire
What divisibility rules would we use if we wanted to test divisibility by:
18
2
and 9 rules
45
5
and 9
36
4 and 9
40
5 and 8
An easy Year 10 Maths Olympiad problem:
Find the smallest positive integer which consists only of 0s and 1s, and which is divisible by 12.
Since in must be divisible by 4, the only possibility for the last two digits is 00.
It must have at least three 1s to be divisible by 3 (as we can’t have zero 0s).Therefore 11100 is the answer.
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Exercise 5
Problem sheet of Junior and Intermediate Olympiad problems.
Work in pairs/groups if you wish.
Answers on next slides.
(See printout)Slide40
Question 1
[J31] Every digit of a given positive integer is either a 3 or a 4 with each occurring at least once. The integer is divisible by both 3 and 4. What is the smallest such integer?
?Slide41
Question 2
[J50] The eight-digit number “
ppppqqqq
”, where p and q are digits, is a multiple of 45. What are the possible values of p?
?Slide42
Question 3
[M07] (a) A positive integer N is written using only the digits 2 and 3, with each appearing at least once. If N is divisible by 2 and by 3, what is the smallest possible integer N
?
(b) A positive integer M is written using only the digits 8 and 9, with each appearing at least once. If M is divisible by 8 and by 9, what is the smallest possible integer M?
?
?Slide43
Question 4
[M55] A palindromic number is one which reads the same when its digits are reversed, for example 23832. What is the largest six-digit palindromic number which is exactly divisible by 15?
?Slide44
Question 5
[J16] Find a rule which predicts exactly when five consecutive integers have sum divisible by 15.
?Slide45
Question 6
[M96] Find the possible values of the digits p and q, given that the five-digit number ‘p543q’ is a multiple of 36.
?Slide46
Question 7
[M127] The five-digit number ‘
’, where
and
are digits, is divisible by 36. Find all possible such five-digit numbers.
?Slide47
Question 8
[M31] Find the smallest positive multiple of 35 whose digits are all the same as each other.
?Slide48
Question 9
Show that:
is divisible by 6 for all integers
.
is divisible by 24 for all integers
.
This is the product of three consecutive numbers. One of the three numbers must be divisible by 3, so the product is divisible by 3. Similarly, at least one of the three numbers is divisible by 2, so the product is divisible by 2. Therefore, the product is divisible by 6.
This is the product of 4 consecutive numbers.
At least one of the four is divisible by 3.
Exactly two of the numbers will be divisible by 2.
However, one of the four numbers will be divisible by 4, giving an extra factor of 2. Overall, this means the product is divisible by
.
?Slide49
Question 10
[Based on
NRich
] If the digits 5, 6, 7 and 8 are inserted at random in 3_1_4_0_92 (one in each
space),
what is the probability that the number created will be a multiple of
396 if:
Each of 5, 6, 7, 8 is used exactly once in each of the four gaps.
Each of 5, 6, 7, 8 can be used multiple times.
[JAF solution] If number is divisible by 396, it is divisible by
.
If we used one each of 5, 6, 7, 8 to fill the gaps, all three divisibility rules, for 4, 9 and 11, would be satisfied regardless of order, since the last two digits are fixed and all inserted digits are in even positions. Thus the probability is 1.b) If we used 5, 6, 7, 8, there are
possible orderings.However, we could also use 5, 5, 8, 8 to fill the gaps, as this will not affect the digit sums involved in either the divisibility by 9 or 11 rules (since all digits are inserted in even positions), and the last two digits are fixed for the purposes of the 4 rule. There are 6 possible orderings of these digits.6, 6, 7, 7 is also possible, which by the same reasoning, gives 6 possible orderings.5, 7, 7, 7 is also possible, which gives 4 orderings.6, 6, 6, 8 is also possible, which gives 4 orderings.
There are therefore 4 + 4 + 6 + 6 + 24 = 44 possible numbers divisible by 396.However, there are
total possible ways of using 5, 6, 7 or 8 for each of the four gaps.
The probability is therefore
?Slide50
Year 7 Divisibility Within EquationsSlide51
STARTER
:: Divisibility of Expressions
Given that
is an integer, for each expression, identify whether the statement is always true, always false, or sometimes true.
Always
true
Always
false
Sometimes true
is divisible by 5.
(5 times a number is clearly
in the 5 times table)
is divisible
by 4.
(This will always be one more than a multiple of 4)
is divisible
by 4.
is divisible
by 2.
is divisible by 2.
N
is divisible by 2.
(one
of
and
is even, and
)
N
is a square
(The product of two squares is a square
)
Always
true
Always
false
Sometimes true
(5 times a number is clearly
in the 5 times table)
(This will always be one more than a multiple of 4)
(The product of two squares is a square
)
?
?
?
?
?
?
?Slide52
Factors within equations
What can you tell about the following numbers in these equations?
?
?
What can you therefore say about
in this equation, if
and
are integers?
Since
and 150 are divisible by 5,
must be.
What therefore can you say about
?
4 is not divisible by 5. Therefore
must be.
?
?Slide53
If
then
If
then
Divisibility in Equations
Find
positive
integer
solutions to:
One strategy would be to try different values of
and see if it works for
. e.g. If
:
would be 45. But 45 doesn’t divide by 4.
Can you think of a more intelligent way?
and 50 are multiples of 5, therefore
is a multiple of 5, and therefore
is a multiple of 5.
Therefore
can only be:
If
was 15 then
, then
would have to be negative.
?
?
?
?
?
Bro Tip
: It’s sometimes convenient to write the solutions as
Bro Side Note
: The posh name for this is a ‘
linear Diophantine equation
’. The ‘Diophantine’ bit means we’re only looking for integer solutions. The ‘linear’ bit means that if we plotted the solutions
on some axes, they’d form a straight line.
Slide54
More examples
Find positive integer solutions to:
As
and 90 are divisible by 6,
needs to be divisible by 6, therefore
is divisible by 6.
can be 6 or 12 (as next value of
is 18, and
would be greater than 90).
If
If
Does the following equation have any integer solutions?
No. The left-hand-side of the equation is odd but the right-hand-side of the equation is even.
?
Bro Note
: What is good about this method is that we have not only found all the solutions: we’ve shown there can’t be any others. This is very important in maths!
?Slide55
Test Your Understanding
[JMC 2012 Q25] The interior angles of a triangle are
,
and
, where
are positive integers. What is the value of
?
must be divisible by 13. 13 is the only possibility.
If
. Therefore
Harder One:
?
[JMC 2010 Q12] Sir Lance has a lot of tables and chairs in his house. Each rectangular table seats eight people and each round table seats five people. What is the smallest number of tables he will need to use to seat 35 guests
and himself
, without any of the seating around these tables remaining unoccupied?
(Hint: if
is the number of rectangular tables, and
is number of circular tables, form an equation first that looks like what we’ve previously seen)
A 4 B
5
C
6
D
7
E
8
is divisible by 4 therefore
is. When
. So answer is C
Easier One:
?Slide56
Exercise 6
Find all positive integer solutions to the following:
a)
b)
c)
d)
[JMC 2005 Q16] ‘Saturn’ chocolate bars are packed either in boxes of 5 or boxes of 12. What is the smallest number of full boxes required to pack exactly 2005 ‘Saturn’ bars?
A 118 B 167 C 168
D 170 E 401
Solution: D
[
JMO 2012 A4] A book costs £3.40 and a magazine costs £1.60. Clara spends exactly £23 on books and magazines. How many magazines does she buy
?
Solution: 8
[IMC 2001 Q16] The Pythagoras Patisseries sells triangular cakes at 39p each and square buns at 23p each. For her party, Helen spent exactly £5.12 on an assortment of these cakes and buns. How many items in total did she buy?
A 15
B
16
C
17
D
18
E 19
Solution:
B
If
is divisible by 3, what can you say about?
a)
?
Divisible by 9
b)
?
Divisible by 36
c)
?
Divisible by 4
c)
?
Divisible by 18
1
2
3
4
5
?
?
?
?
?
?
?
?
?
?
?Slide57
Fini
(any slides after this are supplementary)Slide58
BONUS LESSON! Use #3
: Trailing zeroes
Can you think of a rule that tells us the number of trailing zeroes from the prime factorisation? Why do you think it works?
It is the lowest power of the 2 and 5. This is because each 2-5 pair forms a factor of 10, which puts a 0 on the end of the number. The lower power tells us how many pairs we can make.
Prime Factorisation
Number
Trailing Zeroes
15 000
3
5600
2
48 400 000
5
Prime Factorisation
Number
Trailing Zeroes
15 000
3
5600
2
48 400 000
5
?Slide59
Examples
How many zeroes are on the end of
?
80 trailing zeroes.
8!
i
s said “8 factorial” and means
How many trailing zeroes does 8! have?
Thus there is only 1 trailing zero. Note that
seems to give us a lot more twos than fives, so we only need to count the number of fives to get that trailing zeroes.
Without using your calculator, what is the last non-zero digit of
?
If we get rid of all the trailing zeroes, the last digit will be the one we want. We get rid of the five 2-5 pairs to leave
. This is 36, so the last non-zero digit is 6.
Q
Q
Q
?
?
?Slide60
Test Your Understanding
How many zeroes are on the end of
?
40 trailing zeroes.
What is the last non-zero digit of
?
Getting rid of trailing zeroes leaves
, so last non-zero digit is 4.
How many zeroes are at the end of 25! (remember that we only care about the number of 5s in the prime factorisation)
Each multiple of 5 in 25 x 24 x … gives us a prime factor of 5, and there’s 5 of them. However 25 gives us an extra 5. So we have six 5s, and lots of 2s, giving six trailing zeroes.
a
b
N
?
?
?Slide61
Exercise 7
(on provided sheet)
How many zeroes are at the end of
?
Solution: 31
How many zeroes are at the end of
?
Solution: 20
What
is the last non-zero digit of:
Solution: 5
Solution: 8
Solution: 2
[
JMO 2010 A3] Tom correctly works out
and writes down his answer in full. How many digits does he write down in his full answer
?
Solution: 11
[IMC 2007 Q7] If the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 are all multiplied together, how many zeros are at the end of the answer
?
Solution: 2
[IMC 2000 Q18] The number
is written out in full. How many zeroes are there at the end of the number
?
Solution: 6
[Kangaroo Pink 2012 Q16] What is the last non-zero digit when
is evaluated
?
Solution: 6
[Kangaroo Grey 2004 Q25] The number
is the product of the first 100 positive whole numbers. If all the digits of
were written out, what digit would be next to all the zeros at the end?
A 2
B
4 C 6 D 8
E 9
Solution: 4
[SMC 2001 Q15] Sam correctly calculates the value of
. How many digits does her answer contain
?
Solution: 11
[Senior Kangaroo 2012 Q1] How many zeroes are there at the end of the number which is the product of the first 2012 prime numbers
?
Solution: 1
Find two integers, neither of which has a zero digit, whose product is 1 000 000
.
1
2
3
4
5
6
7
8
9
10
11
?
?
?
?
?
?
?
?
?
?
?
?
?Slide62
Supplementary Slides on Divisibility
“
is divisible by 15
”.
Try different values of
to see for what values this is true.
For example, if
, then
is 20, but 20 is not divisible by 15.
What property does
have when it works?
must be divisible by 3
.
?Slide63
Thinking in buckets
“
is divisible by 2”
2
Click for
Bromanimation
Factors
The 3 bucket can’t have the factor of 2, so it must have been in the
bucket.
Therefore
is divisible by 2.
We can think of the
and
of
as ‘buckets’ which we can put prime factors in if it is a factor of the number/expression.
Slide64
Thinking in buckets
“
is divisible by 3”
3
Click for
Bropossibility
1
Factors
The 3 may have come from the 6 bucket (6 is divisible by 3).
Click for
Bropossibility
2
But it may or may not have come from the
bucket (depending on whether
is divisible by 3)
Because the prime factor of 3 could have just come from the 6, we can’t guarantee anything about the divisibility of
Slide65
Thinking in buckets
“
is divisible by 6”
3
Click for
Bropossibility
1
Factors
We have a factor of 2 and 3 to put in the buckets. The 2 must have come from the
bucket, but the 3 could have come from the ‘3 bucket’. In such a case, we can guarantee
is divisible by 2.
2
Click for
Bropossibility
2
But the 3 might have also been in the
bucket.
would be divisible by 6.
There what can we guarantee about
?
It must be divisible by 2, but we can’t guarantee it will be divisible by 6 (because of the first case).
?
?Slide66
Thinking in buckets
“
is divisible by 3”
3
Click for
Bromanimation
Factors
This one’s more complicated. The 3 has to go in one of the buckets. But the two buckets represent the same number (
), so we know both buckets must have a 3.
We know therefore:
is divisible by 3.
is divisible by 9.
3Slide67
Card Sort
Match the statements with the
strongest
statement they MUST result in.
(By strongest, I mean for example that “is divisible by 8” is stronger than “is divisible by 2” as it is more restrictive)
Some cards may not be used and some orange cards may match multiple green.
is divisible by 5.
is divisible by 2.
is divisible by 5.
is divisible by 5.
We don’t know anything about
is divisible by 10.
is divisible by 5
is divisible by 3.
is divisible by 6.
is divisible by 4.
is divisible by 4
is divisible by 6
is divisible by 15.
is divisible by 25.
is divisible by 27.
is divisible by 24
Slide68
Further Divisibility of Expressions
is divisible by 2. What is the largest number we can say
is divisible by?
Try a few values of
first and see.
Factors
2
2
2
If we add a multiple of 4 and a multiple of 2, it will give a multiple of 2 but not of 4.
Answer:
8
?Slide69
More Examples
If
is divisible by 3, what can we say about the divisibility of…
and
are in the 3 times table, but
will be two more than a multiple of 3.
So the expression is divisible by
.
We get a factor of 3 three times from
.
is not divisible by 3, but is even, giving us a factor of 2.
So the expression is divisible by
.
Test Your Understanding:
?
?
a
b