Factors and Multiples

Learning outcome

List factors without omissions, generate multiples and identify common factors and common positive multiples.

Concepts and reasons

For positive integers, a is a factor or divisor of n when n = a × b for some positive integer b. The same statement says that n is a multiple of a. These terms describe opposite directions of the same exact-division relationship.

Every positive integer has factors 1 and itself. Its positive factors are finite because none exceeds the number. Its positive multiples are n, 2n, 3n, …, continuing without end.

Zero is also a multiple of every positive integer because 0 = n × 0, but it is excluded when positive multiples are requested. Zero cannot be used as a divisor; division by zero is undefined.

Find factors in pairs. Whenever a divides n, its partner is n/a. Test candidates until a × a > n, since later partners have already appeared. For a perfect square, the square-root pair repeats the same factor, so write it only once.

Common factors divide each number in a group. The largest is the highest common factor, or HCF. Common multiples are multiples of every number in the group. The smallest positive one is the least common multiple, or LCM. “Positive” matters: zero is not the LCM under this convention.

Worked examples

Example 1. For 28, the factor pairs are 1 × 28, 2 × 14 and 4 × 7. Therefore its positive factors are 1, 2, 4, 7, 14, 28. Candidates 3 and 5 fail; 6 × 6 already exceeds 28.

Example 2. Factors of 18 are 1, 2, 3, 6, 9, 18. Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Their common factors are 1, 2, 3, 6, so HCF = 6.

Example 3. Positive multiples of 4 begin 4, 8, 12; those of 6 begin 6, 12. The first shared value is 12, so LCM = 12.

Common traps

Do not swap factors and multiples. A number is both its own factor and its own multiple. Do not count a square-root factor twice or include zero among positive multiples.

Practice questions

1. List all positive factors of 36.
2. Write the first four positive multiples of 7.
3. Find the HCF and LCM of 8 and 12.

Answers and explanations

1. The pairs are 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. The distinct factors are 1, 2, 3, 4, 6, 9, 12, 18, 36.
2. They are 7, 14, 21, 28, obtained by multiplying 7 by 1, 2, 3, 4.
3. Common factors are 1, 2, 4, giving HCF = 4. Multiples begin 8, 16, 24 and 12, 24, giving LCM = 24.