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All courses Math Foundations · MF.NS.5 5 of 60
Find factors, multiples, common factors, and common multiples to solve numerical and contextual problems.

List all factors of a number

Problem
List all positive factors of \(36\) from least to greatest.
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Hint

Begin by finding factor pairs of 36, starting with 1 × 36. Then test the next whole numbers to see which ones divide 36 evenly.

Factors come in multiplication pairs whose product is 36. Be sure to record both numbers in each pair and stop when the pairs start repeating.

Solution walkthrough

01

Use factor pairs

\[\text{factor}~\text{pair}~\text{product}~=~36\]

A positive factor of 36 appears in a whole-number pair whose product is 36. Listing pairs prevents omissions.

02

List the factor pairs

\[1~x~36,~2~x~18,~3~x~12,~4~x~9,~6~x~6\]

Testing divisors in order gives every positive factor pair. After 6 × 6, the pairs would repeat in reverse order.

03

Collect the distinct factors

\[1,~2,~3,~4,~6,~9,~12,~18,~36\]

Taking both numbers from every pair and arranging them from least to greatest produces the complete factor list. The repeated 6 from 6 × 6 is written only once.

04

Check completeness

\[9~\text{factors};~1~x~36,~2~x~18,~3~x~12,~4~x~9,~6~x~6\]

Every listed number participates in one of the factor pairs, and the pair search reached the square root 6, so no positive factors are missing.

05

Answer in order

\[1,~2,~3,~4,~6,~9,~12,~18,~36\]

These are all and only the positive whole numbers that divide 36 evenly. This matches the correct choice.

+

Another way

  1. Use 36 = 2² × 3²; products 2a × 3b with exponents 0, 1, or 2 generate the same nine factors.

!

Common mistake

Do not stop after listing divisors below 6. Each small divisor has a larger partner, such as 4 with 9 and 2 with 18.

Solution walkthrough video