1. Skipton Girls’ High School
Number: Number Theory
Skipton Girls’ High School
Objectives: Have an appreciation of properties of integers (whole numbers), including finding the Lowest Common Multiple, Highest Common Factor. Extra: Divisibility rules.

2. For Teacher Use: Recommended lesson structure:
Lesson 1: Introduction to Number Theory/Sums of primes+squares problems.
Lesson 2: Prime Factorisation
Lesson 3: LCM/HCF
Lesson 4: Uses of Prime Factorisations
Lesson 5+6: Divisibility Rules
Extension: Divisibility of Terms/Within Equations
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3. Starter
Key Skill: You should try to memorise these.
List the following numbers in your books.
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, 37
The 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 1+2+3=6.
Find the first perfect number after 6.
Key Fact: All perfect numbers are triangular numbers.
Solution: 𝟐𝟖=𝟏+𝟐+𝟒+𝟕+𝟏𝟒 ?
Other numbers you might want to investigate yourself: Tetrahedric numbers, Fibonacci numbers.

4. ! ! 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. ? ? ? ? ? ? !
Distinct integers:
Numbers which are different!
Key Note: A ‘perfect square’ more generally refers to ‘something squared’, which can be an algebraic expression.
For example 𝑥 is a ‘perfect square’, but is not necessarily a square number, e.g. if 𝑥=0.5 (as =2.25) ?

5. 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, , , , , )
Key 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)
𝟔𝟓=𝟒𝟗+𝟏𝟔=𝟔𝟒+𝟏
(Side note: the smallest square number expressible as the sum of two squares in two different ways is 𝟐 𝟓 𝟐 = 𝟕 𝟐 +𝟐 𝟒 𝟐 =𝟐 𝟎 𝟐 +𝟏 𝟓 𝟐 ) ?

6. Exercise 1 ? ? ? (Problems on provided sheet)
[JMC 2015 Q11] What is the smallest prime number that is the sum of three different prime numbers? A B C D E 23 Solution: D
[JMO 1999 A2] In how many different ways can 50 be written as the sum of two prime numbers? (Note: 𝑥+𝑦 and 𝑦+𝑥 do not count as different.) Solution: 4 ways (𝟒𝟕+𝟑, 𝟒𝟑+𝟕, 𝟑𝟕+𝟏𝟑, 𝟑𝟏+𝟏𝟗)
[JMO 2009 A3] The positive whole numbers 𝑎, 𝑏 and 𝑐 are all different and 𝑎 2 + 𝑏 2 + 𝑐 2 =121. What is the value of 𝑎+𝑏+𝑐?
Solution: 17
Key Tip: Use your lists of numbers from the starter. 1 ? 2 ? 3 ?

7. Exercise 1
[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 B C D E Solution: 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 C D E Solution: E
[JMC 2010 Q22] Kiran writes down six different prime numbers, 𝑝, 𝑞, 𝑟, 𝑠, 𝑡, 𝑢, all less than 20, such that 𝑝+𝑞=𝑟+𝑠=𝑡+𝑢. What is the value of 𝑝+𝑞? A B C D E Solution: E
[TMC Regional 2009 Q9] can be expressed as the sum of two primes in exactly one way. What is the larger of the two primes? Solution: Note that odd = odd + even only. Thus one of the two primes must be 2. 4 ? 5
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. ? 6 ? 7 ?

8. Exercise 1
[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? N ? N ?

9. Prime Factorisation 30=2×3×5 12= 2 2 ×3 120 20 6 4 ? ? ? = 2 3 ×3×5 5
To find the prime factorisation of a number is to express it as a product of prime numbers. ?
30=2×3×5 12= 2 2 ×3 ?
Key Tip: While 12=2×2×3 is also correct, we can use ‘index notation’ to group prime factors together that are the same. ?
120
= 2 3 ×3×5
We can use a ‘tree’ to help us with the working.
For each number, find two numbers the multiply to give it. 20 6 4 5 2 3
If you get to a prime, we can’t branch out further, so we have a ‘leaf’. It’s helpful to circle the leaves. 2 2

10. 2250=2× 3 2 × 5 3 Another quick example 2250 225 10 45 9 ?
Possible Tree ?
2250
225 10 45 5 2 5 9 5 3 3

11. 1350=2× 3 3 × 5 2 Check Your Understanding ? Some Possible Trees ?
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=2× 3 3 × 5 2 ?
Some Possible Trees ?
1350
1350 5
270 10
135 90 3 2 5 5 27 3 3 30 9 5 6 3 3 2 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.

12. 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?
10 3 = 2× =2×5×2×5×2× = 2 3 × 5 3 ?
2 3 × 2 4 = 2×2×2 × 2×2×2×2 = 2 7 ? ? ? ?
Bro Note: This is an example of a ‘law of indices’, which you will learn more about in Year 8.
Quickfire Questions: N:
9 10 = 𝟑 𝟐𝟎
8 10 = 𝟐 𝟑𝟎
= 𝟐 𝟏𝟎𝟎 × 𝟓 𝟏𝟎𝟎
12 12 = 𝟐 𝟐𝟒 × 𝟑 𝟏𝟐
1 8 5 = 𝟐 𝟓 ×𝟑 𝟏𝟎
12× =𝟑× 𝟐 𝟏𝟎𝟐
Working: = 3 10 × 3 10 = 3 20 ?
6 5 = 𝟐 𝟓 × 𝟑 𝟓
= 𝟑 𝟏𝟎𝟎 × 𝟕 𝟏𝟎𝟎 = 𝟑 𝟏𝟓 × 𝟓 𝟏𝟓
70 4 = 𝟐 𝟒 × 𝟓 𝟒 × 𝟕 𝟒
5 5 × 5 2 = 𝟓 𝟕
7 20 × 7 20 = 𝟕 𝟒𝟎 ? ? ? ? ? ? ? ? ? ? ?

13. Exercise 2
By drawing a tree of otherwise, find prime factorisations (in index form) for the following numbers.
28= 𝟐 𝟐 ×𝟕 75=𝟑× 𝟓 𝟐 1000= 𝟐 𝟑 × 𝟓 𝟑 378=𝟐× 𝟑 𝟑 ×𝟕 396= 𝟐 𝟐 × 𝟑 𝟐 ×𝟏𝟏 1755= 𝟑 𝟐 × 𝟓 𝟐 ×𝟏𝟑 432= 𝟐 𝟒 × 𝟑 𝟑
Is odd or even?
An odd number to any power is always odd.
Put in prime factorised form: 3 5 × 3 6 = 𝟑 𝟏𝟏 × 3 3 × 2 4 × 3 5 = 𝟐 𝟔 × 𝟑 𝟖 = 𝟐 𝟕 × 𝟕 𝟕 = 𝟓 𝟖 ×𝟕 𝟑 𝟖 = 𝟓 𝟓𝟓 × 𝟏𝟏 𝟓𝟓 6× 3 20 =𝟐× 𝟑 𝟐𝟏 1 5
Prime factorise the following:
10 000= 𝟐 𝟒 × 𝟓 𝟒 = 𝟓 𝟏𝟒 = 𝟐 𝟑𝟎 × 𝟑 𝟔𝟎 × 8 6 = 𝟐 𝟑𝟎 = 𝟐 𝟔𝟎𝟎 × 𝟓 𝟔𝟎𝟎
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 𝟐×𝟑 or 𝟐×𝟑×𝟏 or 𝟐×𝟑×𝟏×𝟏. 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 ? ? ? ? ? ? ? ? ? ? N1 ? ? 2 ? ? 3 ? ? ? N2 ? ? ? ? 4
What are the factors of 2 5 ? Give your answers in index form. 𝟏, 𝟐 𝟏 , 𝟐 𝟐 , 𝟐 𝟑 , 𝟐 𝟒 , 𝟐 𝟓 ?

14. Starter
List the factors of 3 5
(keeping your factors in prime factorised form)
3 5 divides for example by because = 3×3×3×3×3 3×3 =3×3×3= 3 3
You will learn in Year 8 that 3 0 =1, so this is consistent with the pattern.
1, 3 1 , 3 2 , 3 3 , 3 4 , 3 5 ?
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)
3 5 , 3 6 , 3 7 , … ?
One number will be a multiple of another if the prime factor(s) are the same but the powers are greater (or equal).

15. 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.

16. Check Your Understanding?
Key Shortcut: Any multiple of 60 ends with a 0. Therefore the multiple of 72 must be x5, x10, …
𝐿𝐶𝑀 60,72 =𝟑𝟔𝟎 𝐻𝐶𝐹 60,72 =𝟏𝟐 ?
Key Shortcut: Any number which goes into 60 and 72 must also go into their difference! (i.e. 12) ?
𝐿𝐶𝑀 12,21 =𝟖𝟒 𝐻𝐶𝐹 12,21 =𝟑 ?

17. But what about bigger numbers?
792, 378
Sometimes it’s not practical to use this method.
Can we use the prime factorisation somehow? ?
792= 2 3 × 3 2 ×11 378=2× 3 3 ×7 ?

18. But what about bigger numbers?
792= 2 3 × 3 2 ×11 378=2× 3 3 ×7
Method: ‘What wins what loses’
Alternative: Venn Diagram Method
792= 2 3 × ×11 378=2 × 3 3 ×7
792
378 2 3 3
Step 1: Align numbers so that each prime factor has its own column. 11 3 2 2
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 𝟐 𝟑 . In this case 2.
We see what ‘loses’ in each column (where ‘nothing’ always loses against ‘something’)
𝐻𝐶𝐹 792,378 =𝟐× 𝟑 𝟐 =𝟏𝟐 7
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. 𝟐×𝟑×𝟑=𝟏𝟖
For LCM, what is both a multiple of 2 and 2 3 ? Again, from earlier, they both go into 2 3 , i.e. the one that ‘wins’. Repeating for the other numbers:
𝐿𝐶𝑀 792,378 = 𝟐 𝟑 × 𝟑 𝟑 ×𝟕×𝟏𝟏=𝟓𝟓𝟒𝟒 ?
LCM is all numbers multiplied.
𝟐 𝟑 × 𝟑 𝟑 ×𝟕×𝟏𝟏=𝟓𝟓𝟒𝟒 ?

19. More Examples
672= 2 5 × 3 2 ×7 72 = 2 3 × 3 2
588= 2 2 ×3× = 2 4 ×7×11 ? ?
𝑳𝑪𝑴 𝟔𝟕𝟐,𝟕𝟐 = 𝟐 𝟓 × 𝟑 𝟐 ×𝟕=𝟐𝟎𝟏𝟔 𝑯𝑪𝑭 𝟔𝟕𝟐,𝟕𝟐 = 𝟐 𝟑 ×𝟑=𝟐𝟒
(note that if there’s a ‘draw’, both win and both lose)
Line numbers up:
𝟓𝟖𝟖 = 𝟐 𝟐 ×𝟑× 𝟕 𝟐 𝟏𝟐𝟑𝟐= 𝟐 𝟒 ×𝟕×𝟏𝟏
𝑳𝑪𝑴 𝟓𝟖𝟖,𝟏𝟐𝟑𝟐 = 𝟐 𝟒 ×𝟑× 𝟕 𝟐 ×𝟏𝟏=𝟐𝟓𝟖𝟕𝟐 𝑯𝑪𝑭 𝟓𝟖𝟖,𝟏𝟐𝟑𝟐 = 𝟐 𝟐 ×𝟕=𝟐𝟖

20. Check Your Understanding?
1936= 𝟐 𝟒 × 𝟏𝟏 𝟐 792= 𝟐 𝟑 × 𝟑 𝟐 ×𝟏𝟏 ?
𝐿𝐶𝑀= 𝟐 𝟒 × 𝟑 𝟐 × 𝟏𝟏 𝟐 =𝟏𝟕𝟒𝟐𝟒 𝐻𝐶𝐹= 𝟐 𝟑 ×𝟏𝟏=𝟖𝟖 ? ?
If you finish… ?
3675=𝟑× 𝟓 𝟐 × 𝟕 𝟐 875= 𝟓 𝟑 ×𝟕 ? ?
𝐿𝐶𝑀=𝟑× 𝟓 𝟑 × 𝟕 𝟐 =𝟏𝟖𝟑𝟕𝟓 𝐻𝐶𝐹= 𝟓 𝟐 ×𝟕=𝟏𝟕𝟓 ?

21. Exercise 3 N
[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
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 = 105
The 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?
9:36am
Find the LCM and HCF of the following pairs, by prime factorising the numbers first.
a) 36 and 378 LCM = 756, HCF = 18 b) 315 and 3675 LCM = 11025, HCF = 105
c) 72 and 66 LCM = 792, HCF = 6
d) 2880 and 792 LCM = 31680, HCF = 72
e) 375 and 325 LCM = 4875, HCF = 25
f) 252 and 2079 LCM = 8316, HCF = 63 1 ? ? ? ? ? 2 ? 3 ? ? ? ? ? ?

22. Number: Divisibility Rules

23. Divisibility Rules
How can we tell if a number is divisible by: ! 2
Last digit is even. 3
Digits add up to multiple of 3. e.g: 1692: = 18  4
Last two digits are divisible by 4. e.g 5
Last 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: > 246 – 8 = 238 -> 23 – 16 = 7. 8
Last three digits divisible by 8. 9
Digits add up to multiple of 9. 10
Last digit 0. 11
When you sum odd-positioned digits and subtract even-positioned digits, the result is divisible by 11.
e.g : ( ) – (7 + 4) = 22 – 11 = 11, which is divisible by 11. 12
Number divisible by 3 and by 4. ? ? ? ? ? ? ? ? ? ? ?

24. Quickfire Divisibility 4 6 7 9 11 726  168 9196 252 1001 91 216 87912? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

25. 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
131
781
751
221
No! (13×17) ? ? ? ? ? ? ? ? ? ?
Advanced: 13 trick is “Quadruple last digit and add to remaining number. Is result divisible by 13? ? ? ? ?
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 𝟐𝟐𝟏 =𝟏𝟒.𝟖𝟕 because all composite numbers have a factor (other than 1) up to the square root. ?

26. Test Your Understanding
Easier One:
[JMO 1997 A5] Precisely, one of the numbers 234, 2345, 23456, , , is a prime number. Which one must it be? Solution: ?
Harder One:
[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 Solution: A ?