Mathematics notes, chapter 2

Factors and multiples

Textbook Mathematics Form 1 (DLP), pages 30–44

What this chapter covers

List the factors of a whole number and make generalisations about factors (2.1.1)
List the prime factors of a whole number and write it in prime factorisation form (2.1.2)
Explain and find the common factors of whole numbers (2.1.3)
Find the HCF of two or three whole numbers by listing, repeated division and prime factorisation (2.1.4)
Solve problems involving HCF (2.1.5)
Explain and find the common multiples of whole numbers (2.2.1)
Find the LCM of two or three whole numbers by listing, repeated division and prime factorisation (2.2.2)
Solve problems involving LCM (2.2.3)

2.1Factors, prime factors and highest common factor (HCF)

What are factors?

Factors of a number are whole numbers that can divide the number completely (with no remainder).

12 ÷ 1 = 12, 12 ÷ 2 = 6, 12 ÷ 3 = 4, 12 ÷ 4 = 3, 12 ÷ 6 = 2 and 12 ÷ 12 = 1. So the factors of 12 are 1, 2, 3, 4, 6 and 12. You can see the factors by arranging 12 counters in equal rows: every rectangle gives a factor pair.

12 counters arranged as 1 row of 12, 2 rows of 6 and 3 rows of 4 — Each arrangement of 12 counters gives a factor pair

Example 1: is it a factor? (textbook p. 32)

Determine whether (a) 12 is a factor of 36, (b) 9 is a factor of 30.

(a) 36 ÷ 12 = 3, no remainder. So 12 is a factor of 36.
(b) 30 ÷ 9 = 3 remainder 3. So 9 is not a factor of 30.

Example 2: list all the factors of 18 (textbook p. 32)

Write 18 as a product of two numbers, starting from 1: 18 = 1 × 18, 18 = 2 × 9, 18 = 3 × 6.
Try 4 and 5: they do not divide 18 completely. The next number, 6, is already in the list, so stop.

Factors of 18: 1, 2, 3, 6, 9, 18.

Generalisations about factors: 1 is a factor of every number, and every number is a factor of itself. Work in pairs (1 × 18, 2 × 9 …) so you never miss a factor.

(a) List all the factors of 32. (b) Is 14 a factor of 56? Is 18 a factor of 98?

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(a) 32 = 1 × 32 = 2 × 16 = 4 × 8, so the factors are 1, 2, 4, 8, 16, 32.
(b) 56 ÷ 14 = 4, so yes. 98 ÷ 18 = 5 remainder 8, so no.

What are prime factors?

A prime number has exactly two factors, 1 and itself. A prime factor of a number is a factor that is also a prime number.

Numbers 1 to 50 with the prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47 circled — The prime numbers up to 50 (Sieve of Eratosthenes)

Factors of 18 are 1, 2, 3, 6, 9 and 18. Only 2 and 3 are prime, so the prime factors of 18 are 2 and 3.

1 is not a prime number (it has only one factor), so 1 is never a prime factor. Also, 9, 15, 21, 27, 33, 39, 49 and 51 are odd but not prime.

Example 3: prime factors of 84 (textbook p. 33)

Is (a) 2, (b) 3, (c) 5 a prime factor of 84?

(a) 84 ÷ 2 = 42 and 2 is prime: yes.
(b) 84 ÷ 3 = 28 and 3 is prime: yes.
(c) 84 ÷ 5 = 16 remainder 4. 5 is prime but does not divide 84 completely: no.

Example 4: list all the prime factors of 72 (textbook p. 34)

List all the factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72.
Pick out the prime numbers in the list: 2 and 3.

The prime factors of 72 are 2 and 3.

Prime factorisation

Prime factorisation means writing a number as a product of its prime factors, for example 18 = 2 × 3 × 3.

Example 5: express 60 in prime factorisation (textbook p. 34)

Method 1: repeated division. Divide by the smallest prime that works. Keep dividing until the quotient is 1.

60 divided by 2 is 30, by 2 is 15, by 3 is 5, by 5 is 1

Method 2: factor tree. Split the number into two factors again and again until every branch ends in a prime (circled).

Two factor trees of 60: 60 = 4 × 15 = 2 × 2 × 3 × 5, and 60 = 2 × 30, 30 = 3 × 10, 10 = 2 × 5

Both ways give 60 = 2 × 2 × 3 × 5. A different first split still gives the same primes.

Express (a) 96 and (b) 120 in the form of prime factorisation.

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(a) 96 = 2 × 2 × 2 × 2 × 2 × 3
(b) 120 = 2 × 2 × 2 × 3 × 5

Common factors

A common factor of two or more numbers is a number that is a factor of all of them.

Venn diagram. Factors of 18 only: 9, 18. Both: 1, 2, 3, 6. Factors of 24 only: 4, 8, 12, 24. — The overlap holds the common factors of 18 and 24: 1, 2, 3 and 6

Example 6: is it a common factor? (textbook p. 35)

Is 6 a common factor of 24 and 36? 24 ÷ 6 = 4 and 36 ÷ 6 = 6. Yes.
Is 4 a common factor of 30, 40 and 60? 30 ÷ 4 = 7 remainder 2. No — one number is enough to fail.

Example 7: common factors of 20, 44, 56 and 64 (textbook p. 35)

Number	Factors
20	1, 2, 4, 5, 10, 20
44	1, 2, 4, 11, 22, 44
56	1, 2, 4, 7, 8, 14, 28, 56
64	1, 2, 4, 8, 16, 32, 64

Common factors: 1, 2 and 4.

Highest common factor (HCF)

The highest common factor (HCF) is the greatest of the common factors. It is also called the greatest common divisor.

The common factors of 20, 44, 56 and 64 are 1, 2 and 4, so their HCF is 4. There are three methods.

Example 8: three methods for the HCF (textbook p. 36)

(a) Listing the common factors — HCF of 18 and 24.
Factors of 18: 1, 2, 3, 6, 9, 18. Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24.
Common factors: 1, 2, 3, 6. HCF = 6.

(b) Repeated division — HCF of 36, 60 and 72. Divide by a prime that divides all the numbers. Stop when no prime divides them all. Multiply the divisors only.

2 into 36, 60, 72 gives 18, 30, 36; 2 gives 9, 15, 18; 3 gives 3, 5, 6. HCF = 2 × 2 × 3 = 12

(c) Prime factorisation — HCF of 48, 64 and 80.

48	= 2 × 2 × 2 × 2 × 3
64	= 2 × 2 × 2 × 2 × 2 × 2
80	= 2 × 2 × 2 × 2 × 5

Multiply the prime factors common to all three: HCF = 2 × 2 × 2 × 2 = 16.

In the repeated division for the HCF, multiply only the divisors on the left. Do not multiply the numbers left at the bottom (3, 5, 6) — that would give a much bigger number, not the HCF.

Find the HCF of (a) 30 and 72, (b) 27, 45 and 60.

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(a) 30 = 2 × 3 × 5, 72 = 2 × 2 × 2 × 3 × 3. HCF = 2 × 3 = 6
(b) 27 = 3 × 3 × 3, 45 = 3 × 3 × 5, 60 = 2 × 2 × 3 × 5. HCF = 3

Solving problems involving HCF

Look for the words maximum number of packs / groups / plates or biggest size when things are shared equally or cut with nothing left over. These need the HCF.

Example (textbook p. 37): Boy Scouts' charity packs

252 shirts, 180 pairs of trousers and 108 pairs of socks are divided equally into packs. What is the maximum number of packs?

Understand: every pack gets the same of each item; find the most packs.
Plan: find the HCF of 252, 180 and 108.
Solve: 252 = 2 × 2 × 3 × 3 × 7, 180 = 2 × 2 × 3 × 3 × 5, 108 = 2 × 2 × 3 × 3 × 3. HCF = 2 × 2 × 3 × 3 = 36 packs.
Check: 252 ÷ 36 = 7 shirts, 180 ÷ 36 = 5 trousers, 108 ÷ 36 = 3 pairs of socks in each pack.

Worked problem (Mastery Q 2.1, textbook p. 38): cutting paper into squares

Anita cuts paper 260 cm long and 20 cm wide into equal squares with no paper left. What is the biggest possible size of the squares?

The side of the square must divide both 260 and 20 completely.
HCF of 260 and 20 = 20. So the squares are 20 cm × 20 cm.
Check: 260 ÷ 20 = 13 squares in one row.

A strip 260 cm by 20 cm divided by dashed lines into 13 squares of 20 cm

The Police Cadets share 60 sacks of rice, 78 bottles of cooking oil and 90 bottles of mineral water equally into boxes. What is the maximum number of boxes?

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60 = 2 × 2 × 3 × 5, 78 = 2 × 3 × 13, 90 = 2 × 3 × 3 × 5
HCF = 2 × 3 = 6, so 6 boxes (each with 10 sacks of rice, 13 bottles of oil, 15 bottles of water).

2.2Multiples, common multiples and lowest common multiple (LCM)

Multiples and common multiples

Multiples of a number are the products when it is multiplied by 1, 2, 3, … For example 9, 18, 27, … are multiples of 9. A common multiple is a multiple of all the given numbers.

Number line 0 to 50. Jumps of 6 land on 6, 12, 18, 24, 30, 36, 42, 48. Jumps of 8 land on 8, 16, 24, 32, 40, 48. 24 and 48 are ringed. — Jumps of 6 and jumps of 8 first land together at 24

Multiples of 6: 6, 12, 18, 24, 30, … Multiples of 8: 8, 16, 24, 32, 40, … So 24 is a common multiple of 6 and 8, and 24 can be divided completely by both 6 and 8.

All the other common multiples are multiples of the first one. The common multiples of 6 and 8 are 24, 48, 72, 96, … — there are infinitely many.

Example 9: is it a common multiple? (textbook pp. 38–39)

Is 32 a common multiple of 2 and 8? 32 ÷ 2 = 16 and 32 ÷ 8 = 4. Yes.
Is 60 a common multiple of 5, 15 and 24? 60 ÷ 24 = 2 remainder 12. No.
Is 72 a common multiple of 8, 9, 18 and 24? 72 ÷ 8 = 9, 72 ÷ 9 = 8, 72 ÷ 18 = 4, 72 ÷ 24 = 3. Yes.

Example 10: first five common multiples of 2, 3 and 4 (textbook p. 40)

Multiples of 2: 2, 4, 6, 8, 10, 12 … Multiples of 3: 3, 6, 9, 12 … Multiples of 4: 4, 8, 12 …
The first common multiple is 12. Now list multiples of 12.

First five common multiples: 12, 24, 36, 48, 60.

Lowest common multiple (LCM)

The lowest common multiple (LCM) is the smallest of the common multiples. The LCM of 6 and 8 is 24.

Example 11: three methods for the LCM (textbook pp. 40–41)

(a) Listing multiples — LCM of 2 and 3. Multiples of 2: 2, 4, 6, 8 … Multiples of 3: 3, 6, 9 … LCM = 6.

(b) Repeated division — LCM of 3, 6 and 9. Divide by a prime that divides at least one number. Bring down any number that cannot be divided. Continue until every quotient is 1, then multiply all the divisors.

2 into 3, 6, 9 gives 3, 3, 9; 3 gives 1, 1, 3; 3 gives 1, 1, 1. LCM = 2 × 3 × 3 = 18

(c) Prime factorisation — LCM of 3, 8 and 12. 3 = 3, 8 = 2 × 2 × 2, 12 = 2 × 2 × 3. Take each prime the greatest number of times it appears in any one number: three 2s (from 8) and one 3. LCM = 2 × 2 × 2 × 3 = 24.

Shortcut: list multiples of the greatest number (12, 24, 36 …) and stop at the first one that 3 and 8 divide completely: 24.

Do not mix up the two ladders. HCF: divide only when the prime divides all numbers, then stop. LCM: keep going until the bottom row is all 1s. An LCM can never be smaller than the biggest number, and an HCF can never be bigger than the smallest number.

Find the LCM of (a) 10 and 15, (b) 9, 12 and 21.

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(a) 10 = 2 × 5, 15 = 3 × 5. LCM = 2 × 3 × 5 = 30
(b) 9 = 3 × 3, 12 = 2 × 2 × 3, 21 = 3 × 7. LCM = 2 × 2 × 3 × 3 × 7 = 252

Solving problems involving LCM

Look for "together again", "at the same time", "the same number of" or "smallest / minimum number". These need the LCM.

Example (textbook pp. 41–42): canned coffee and tea

Coffee is sold 6 cans per box and tea 9 cans per box. Ainun wants the same number of cans of each. What is the minimum number of boxes of each?

Plan: find the LCM of 6 and 9 for the same number of cans.
Multiples of 6: 6, 12, 18 … Multiples of 9: 9, 18 … LCM = 18 cans.
Coffee: 18 ÷ 6 = 3 boxes. Tea: 18 ÷ 9 = 2 boxes.
Check: 3 × 6 = 18 and 2 × 9 = 18.

Clue in the question	Use	Answer is
Share equally, maximum number of groups, biggest size, cut with nothing left	HCF	smaller than or equal to the numbers
Together again, same time, same number, smallest / minimum amount	LCM	bigger than or equal to the numbers

(a) One neon light blinks every 9 seconds and another every 12 seconds. They blink together when switched on. After how many seconds do they blink together again?
(b) Nurul, Hui Ling and Gopal cut equal ribbons into pieces of 15 cm, 25 cm and 30 cm with nothing left. What is the shortest possible length of each ribbon?

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(a) LCM of 9 and 12 = 2 × 2 × 3 × 3 = 36 seconds
(b) 15 = 3 × 5, 25 = 5 × 5, 30 = 2 × 3 × 5. LCM = 2 × 3 × 5 × 5 = 150 cm

Summary

Remember	Factors	Multiples
Example	Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 18: 1, 2, 3, 6, 9, 18	Multiples of 6: 6, 12, 18, 24, 30, … Multiples of 8: 8, 16, 24, 32, 40, …
Common	Common factors of 12 and 18: 1, 2, 3, 6	Common multiples of 6 and 8: 24, 48, 72, 96, …
Highest / lowest	HCF of 12 and 18 = 6	LCM of 6 and 8 = 24
Prime factorisation	HCF = primes common to all numbers	LCM = every prime, the most times it appears in one number
Problems	Share equally / biggest size	Together again / same number

Content and examples follow Mathematics Form 1 (DLP) textbook, Chapter 2, pp. 30–44.