Chapter 12 · ICSE Class 5 Math

Factors and Multiples

Every number is built from smaller numbers that divide into it evenly. Drag the slider and watch its factors appear, along with whether it's prime or composite.

The Factor Machine
12
Composite
12

Try it: set the slider to 7 — a number with only two factors is prime.

A factor tree breaking 36 down into its prime factors A factor tree showing 36 splitting into 6 and 6, then further into the prime factors 2, 3, 2, and 3 36 6 6 2 3 2 3
12.1

What is a factor?

A factor of a number is any number that divides into it exactly, with no remainder. Every number has at least two factors: 1 and itself.

Everyday example — If you have 12 sweets to share equally, the factors of 12 (1, 2, 3, 4, 6, 12) tell you all the group sizes that share them evenly with none left over. Try setting the Factor Machine above to 12 to see this.
12.2

What is a multiple?

1Multiples of a number

A multiple of a number is what you get by multiplying it by a whole number. Multiples of 4 are: 4, 8, 12, 16, 20...

2Factors vs multiples

Factors of a number are smaller than or equal to it. Multiples of a number are equal to or bigger than it, and go on forever.

12.3

Prime and composite numbers

Prime
2, 3, 5, 7, 11...

Has exactly 2 factors: 1 and itself.

Composite
4, 6, 8, 9, 10...

Has more than 2 factors.

Neither
1

Has only 1 factor, so it's neither prime nor composite.

12.4

Prime factorization

1Breaking a number into primes

Prime factorization means writing a number as a multiplication of only prime numbers. Find the prime factors of 36 using a factor tree: 36 = 2 × 18 = 2 × 2 × 9 = 2 × 2 × 3 × 3.

36 = 2 × 2 × 3 × 3

2Using division method

Divide 60 repeatedly by the smallest prime that fits: 60 ÷ 2 = 30, 30 ÷ 2 = 15, 15 ÷ 3 = 5, 5 ÷ 5 = 1.

60 = 2 × 2 × 3 × 5

Why it matters — Prime factorization is a faster way to find HCF and LCM for bigger numbers, since you only need to compare their prime building blocks instead of listing every factor or multiple.
12.5

Finding LCM and HCF

1HCF — Highest Common Factor

Find the HCF of 12 and 18. Factors of 12: 1,2,3,4,6,12. Factors of 18: 1,2,3,6,9,18.

Common factors: 1,2,3,6. HCF = 6

2LCM — Lowest Common Multiple

Find the LCM of 4 and 6. Multiples of 4: 4,8,12,16. Multiples of 6: 6,12,18.

First common multiple: LCM = 12

3HCF and LCM using prime factorization

Find the HCF and LCM of 12 and 18 using their prime factors: 12 = 2 × 2 × 3, and 18 = 2 × 3 × 3.

HCF uses the common primes: HCF = 2 × 3 = 6. LCM uses every prime at its highest count: LCM = 2 × 2 × 3 × 3 = 36.

Watch out — HCF is always the smallest of the numbers or smaller — never bigger than the smallest number. LCM is always the largest number or bigger — never smaller than the largest number.
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