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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.
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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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,
? 225, 256
121, 144, 169, 196,
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:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66
?
(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)
?
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.
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 + 1 2 is a
‘perfect square’, but is not
necessarily a square number,
e.g. if 𝑥 = 0.5 (as 1.52 = 2.25)
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.
Key Tip: It often
How many ways are there of expressing 100 as the sum of two primes?
helps to write
Solution: 6 (3 + 97, 11 + 89, 17 + 83, 29 + 71, 41 + 59, 47 + 53)
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 𝟐𝟓𝟐 = 𝟕?𝟐 + 𝟐𝟒𝟐 = 𝟐𝟎𝟐 + 𝟏𝟓𝟐)
Exercise 1
(Problems on provided sheet)
1 [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
?
2
[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)
3
[JMO 2009 A3] The positive whole numbers 𝑎, 𝑏 and 𝑐 are all
different and
𝑎2 + 𝑏2 + 𝑐 2 = 121. What is the value of 𝑎 + 𝑏 + 𝑐?
Solution: 17
?
Key Tip: Use
your lists of
numbers from
the starter.
Exercise 1
4
[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 512
Solution: D
5
[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 24
Solution: E
6
?
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.
[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 16
B 18
C 20
D 22
E 24
Solution: E
7
?
?
[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
N
[JMO 2006 A9] The prime number 11 may be written as the
sum of three prime numbers in two different ways: 2 + 2 + 7
and 3 + 3 + 5. 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
?
N
[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?
?
Prime Factorisation
To find the prime factorisation of a number is to express it as a product of
prime numbers.
?5
30 = 2 × 3 ×
12 = 22 × 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.
= 23 × ?3 × 5
120
20
4
2
6
5
2
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.
Another quick example
2250 =
2
?
2×3
×5
2250
225
9
3
5
3
10
Possible Tree
5
5
2
?
45
3
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 = 2 × 33? × 52
1350
1350
10
2
5
135
5
27
5
3
9
3
270
3
90 Trees
Some Possible
?3 30
3
6
5
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.
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?
103 = 2 × 5? 3
= 2 × 5 × 2 ×? 5 × 2 × 5
= 23 × 53 ?
Quickfire Questions:
5
𝟓
?𝟓
6 =𝟐 ×𝟑
21100 = 𝟑𝟏𝟎𝟎 ×? 𝟕𝟏𝟎𝟎
1515 = 𝟑𝟏𝟓 × 𝟓?𝟏𝟓
704 = 𝟐𝟒 × 𝟓𝟒?× 𝟕𝟒
55 × 52 = 𝟓 𝟕 ?
720 × 720 = 𝟕𝟒𝟎?
2 3 × 24
= 2 × 2 × 2 × ?2 × 2 × 2 × 2
= 27 ?
Bro Note: This is an example of a ‘law
of indices’, which you will learn more
about in Year 8.
N:
Working: 910 = 310 × 310 = 320
910 = 𝟑𝟐𝟎 ?
810 = 𝟐𝟑𝟎 ?
10050 = 𝟐𝟏𝟎𝟎 ×? 𝟓𝟏𝟎𝟎
1212 = 𝟐𝟐𝟒 × ?𝟑𝟏𝟐
185 = 𝟐𝟓 × 𝟑?𝟏𝟎
12 × 2100 = 𝟑 × 𝟐𝟏𝟎𝟐
?
Exercise 2
1
By drawing a tree of otherwise, find prime
factorisations (in index form) for the following
numbers.
28 = 𝟐𝟐 × 𝟕
75 = 𝟑 × 𝟓𝟐
1000 = 𝟐𝟑 × 𝟓𝟑
378 = 𝟐 × 𝟑𝟑 × 𝟕
396 = 𝟐𝟐 × 𝟑𝟐 × 𝟏𝟏
1755 = 𝟑𝟐 × 𝟓𝟐 × 𝟏𝟑
432 = 𝟐𝟒 × 𝟑𝟑
?
?
?
?
?
?
?
2
Is 137776 odd or even?
An odd number to any power is always odd.
3
Put in prime factorised form:
35 × 36 = 𝟑𝟏𝟏
22 × 33 × 24 × 35 = 𝟐 𝟔 × 𝟑 𝟖
147 = 𝟐𝟕 × 𝟕𝟕
3658 = 𝟓𝟖 × 𝟕𝟑𝟖
5555 = 𝟓𝟓𝟓 × 𝟏𝟏𝟓𝟓
6 × 320 = 𝟐 × 𝟑𝟐𝟏
?
?
?
?
?
What are the factors of 25 ? Give your
answers in index form. 𝟏, 𝟐𝟏 , 𝟐𝟐 ,?
𝟐𝟑 , 𝟐𝟒 , 𝟐𝟓
Prime factorise the following:
10 000 = 𝟐𝟒 × 𝟓𝟒
257 = 𝟓𝟏𝟒
1830 = 𝟐𝟑𝟎 × 𝟑𝟔𝟎
46 × 86 = 𝟐𝟑𝟎
1 000 000100 = 𝟐𝟔𝟎𝟎 × 𝟓𝟔𝟎𝟎
?
?
N1
?
?
4
5
?
?
?
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.
?
N2
[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
?
Starter
List the factors of 35
35 divides for example by 32 because
(keeping your factors in prime factorised form)
You will learn in Year 8 that 30 = 1,
so this is consistent with the pattern.
35
32
=
3×3×3×3×3
3×3
= 3 × 3 × 3 = 33
1, 31 , 32 , ?33 , 34 , 35
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 35 which only contains prime
factors of 3.
(keeping your multiples in prime factorised form)
35 , 36 , 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).
Lowest Common Multiple/Highest Common Factor
Multiples of 8:
Multiples of 12:
8, 16, 24, ?32, …
12, 24, 36,? …
Lowest Common Multiple of 8 and 12:
24 ?
For small numbers, we can list out
multiples of the larger number until we see
a multiple of the smaller number.
Factors of 8:
Factors of 12:
1, 2, 4, 8 ?
1, 2, 3, 4,?6, 12
Highest Common Factor of 8 and 12:
4?
For small numbers, we can list out factors
of each number and choose the greatest
number which is common.
Check Your Understanding
?
𝐿𝐶𝑀 60,72 = 𝟑𝟔𝟎
𝐻𝐶𝐹 60,72 = 𝟏𝟐?
Key Shortcut: Any multiple
of 60 ends with a 0.
Therefore the multiple of 72
must be x5, x10, …
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 ?
But what about bigger numbers?
792, 378
Sometimes it’s not practical to use this method.
Can we use the prime factorisation somehow?
3
2
2
?
3
792 =
× × 11
3?
378 = 2 × 3 × 7
But what about bigger numbers?
792 = 23 × 32 × 11
378 = 2 × 33 × 7
Method: ‘What wins what loses’
Alternative: Venn Diagram Method
792 = 23 × 32
× 11
378 = 2 × 33 × 7
792
Step 1: Align numbers so that each
prime factor has its own column.
3
For HCF, out of 2 and 2, what factor is common
to both? We saw from earlier that 2 is a factor of 2
and 23 . We effectively find what ‘loses’ out of 2
and 𝟐𝟑 . In this case 2.
We see what ‘loses’ in each column (where
‘nothing’ always loses against ‘something’)
𝐻𝐶𝐹 792,378 = 𝟐 × 𝟑𝟐 = 𝟏𝟐
For LCM, what is both a multiple of 2 and 23 ?
Again, from earlier, they both go into 23 , i.e. the
one that ‘wins’. Repeating for the other
numbers:
𝐿𝐶𝑀 792,378 = 𝟐𝟑 × 𝟑𝟑 × 𝟕 × 𝟏𝟏 = 𝟓𝟓𝟒𝟒
378
2
2
7
3
3
3 11
2
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
LCM is all numbers multiplied.
𝟐𝟑 × 𝟑𝟑 × 𝟕 ?
× 𝟏𝟏 = 𝟓𝟓𝟒𝟒
More Examples
672 = 25 × 32 × 7
72 = 23 × 32
588 = 22 × 3 × 72
1232 = 24 × 7 × 11
𝑳𝑪𝑴 𝟔𝟕𝟐, 𝟕𝟐
= 𝟐𝟓 × 𝟑𝟐 × 𝟕 = 𝟐𝟎𝟏𝟔
𝑯𝑪𝑭 𝟔𝟕𝟐, 𝟕𝟐
= 𝟐𝟑 × 𝟑 = 𝟐𝟒
Line numbers up:
𝟓𝟖𝟖 = 𝟐𝟐 × 𝟑 × 𝟕𝟐
𝟏𝟐𝟑𝟐 = 𝟐𝟒
× 𝟕 × 𝟏𝟏
?
(note that if there’s a ‘draw’,
both win and both lose)
?
𝑳𝑪𝑴 𝟓𝟖𝟖, 𝟏𝟐𝟑𝟐
= 𝟐𝟒 × 𝟑 × 𝟕𝟐 × 𝟏𝟏 = 𝟐𝟓𝟖𝟕𝟐
𝑯𝑪𝑭 𝟓𝟖𝟖, 𝟏𝟐𝟑𝟐
= 𝟐𝟐 × 𝟕 = 𝟐𝟖
Check Your Understanding
?𝟐
1936 = 𝟐𝟒 × 𝟏𝟏
792 = 𝟐𝟑 × 𝟑𝟐? × 𝟏𝟏
? 𝟐 = 𝟏𝟕𝟒𝟐𝟒
𝐿𝐶𝑀 = 𝟐𝟒 × 𝟑𝟐 × 𝟏𝟏
𝐻𝐶𝐹 = 𝟐𝟑 × 𝟏𝟏? = 𝟖𝟖
If you finish…
3675 = 𝟑 × 𝟓𝟐 ×? 𝟕𝟐
875 = 𝟓𝟑 ×?𝟕
𝐿𝐶𝑀 = 𝟑 × 𝟓𝟑 × 𝟕?𝟐 = 𝟏𝟖𝟑𝟕𝟓
𝐻𝐶𝐹 = 𝟓𝟐 × 𝟕 ?= 𝟏𝟕𝟓
Exercise 3
1
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
?
?
?
?
2
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
?
3
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
?
?
?
?
?
?
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
?
Number: Divisibility Rules
Divisibility Rules
How can we tell if a number is divisible by: !
2
3
4
5
6
?
Digits add up to multiple of 3. e.g: ?
1692: 1+6+9+2 = 18 
Last two digits are divisible by 4. e.g.
? 143328
Last digit is 0 or 5.
?
Number is divisible by 2 and 3 (so use
? tests for 2 and 3).
Last digit is even.
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.
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. 47949: (4 + 9 + 9) – (7 + 4) = 22 – 11 = 11, which is divisible by 11.
12
Number divisible by 3 and by 4.
?
Quickfire Divisibility
4
6 7
? ?
?
726
168
? ? ?
9196 ? ? ?
252
? ? ?
?
? ?
1001
?
? ?
91
216
? ? ?
87912 ? ? ?
9
11
? ?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
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
91
101
234567 
131
781
751
221
7
?
11 Is it prime?
No ?
Yes ?
?
?
?
? 
?
?
?
?
?
Yes ?
No! (13? × 17)
No
Yes
No
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.
?
Test Your Understanding
Easier One:
[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?
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 4
Solution: A
?