Factors, Multiples, and Primes

Understanding factors, multiples, and prime numbers is important for the SET 11+ maths papers. These concepts help you simplify fractions, find common denominators, and solve a variety of number problems. This lesson covers each concept in detail with worked examples and practice strategies.

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Factors

A factor of a number is a whole number that divides into it exactly, with no remainder.

Worked Example

Find all the factors of 36.

Work through pairs that multiply to give 36:

• 1 × 36
• 2 × 18
• 3 × 12
• 4 × 9
• 6 × 6

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 (9 factors in total)

Top Tip: Always work in pairs. Start with 1 × the number, then 2, 3, 4... and stop when the pairs start to repeat.

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Common Factors and the Highest Common Factor (HCF)

Common factors are factors shared by two or more numbers. The Highest Common Factor (HCF) is the largest of these.

Worked Example

Find the HCF of 24 and 36.

• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
• Common factors: 1, 2, 3, 4, 6, 12

HCF = 12

Using prime factor trees

Another method is to find the prime factorisation of each number and identify the shared prime factors.

• 24 = 2 × 2 × 2 × 3
• 36 = 2 × 2 × 3 × 3
• Shared: 2 × 2 × 3 = 12

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Multiples

A multiple of a number is the result of multiplying that number by a whole number.

The first six multiples of 7 are: 7, 14, 21, 28, 35, 42.

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Common Multiples and the Lowest Common Multiple (LCM)

Common multiples are multiples shared by two or more numbers. The Lowest Common Multiple (LCM) is the smallest of these.

Worked Example

Find the LCM of 6 and 8.

• Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48...
• Multiples of 8: 8, 16, 24, 32, 40, 48...

LCM = 24

Using prime factors