October 2026 - Factors & Multiples Study Guide

Core ideas to know

Learn the relationships, not a string of isolated definitions. These are the facts that let you reason through unfamiliar wording.

• Factor: a whole number that divides another with no remainder; the factors of 12 are 1, 2, 3, 4, 6 and 12.
• Multiple: a product in a number’s times table; 12, 24 and 36 are multiples of 12.
• Greatest common factor (GCF): the largest factor shared by two numbers, used to reduce a fraction fully.
• Least common multiple (LCM): the smallest positive multiple shared by two numbers, used as an efficient common denominator.
• A prime number has exactly two positive factors, 1 and itself. The number 1 is neither prime nor composite.

A reliable way to answer

Use the same short process every time so the wording of the question does not decide your method for you.

1. Decide whether the question needs something that divides the numbers (a factor) or something the numbers divide into (a multiple).
2. Prime-factorise both numbers when the shared value is not obvious.
3. For a GCF, keep shared prime factors at their lowest powers; for an LCM, keep every prime factor at its highest power.
4. Check by dividing: a GCF must divide both numbers, while both original numbers must divide the LCM.

Distinctions that earn marks

Terms to know

Divisibility

The property of dividing with a remainder of zero.

Prime factorisation

Writing a number as a product of prime numbers.

Composite number

A positive integer with more than two positive factors.

Key points

• Factors divide; multiples are produced by multiplication.
• Use the GCF to simplify and the LCM to make a common denominator.
• Always verify a proposed shared factor or multiple against both original numbers.