Factor Pairs: Guide and Examples

What Is Factor Pairs?

To understand factor combinations, you first need to know about factors. A factor is a whole number that divides exactly into another number. Because multiplication involves two numbers being multiplied together, factors naturally come in pairs.

For example, if you want to make $15$, you can multiply $3$ by $5$. Together, the numbers $3$ and $5$ form a single factor pair for the number $15$.

When to Use It

You use factor pairs whenever you need to divide a quantity into equal groups, arrange items in a rectangular grid, or simplify fractions. They also help differentiate factors vs multiples, which are often confused.

Exploring combinations of factors even leads to fascinating number theory concepts, such as finding perfect numbers, where the sum of a number's factors equals the original number.

Step-by-Step Method

Finding all factor pairs requires a systematic search to ensure you do not miss any combinations.

1. Start with the number $1$. Every whole number pairs with $1$ to make itself.
2. Move up to $2$. Check if the number is even. If it is, divide by $2$ to find its partner.
3. Test the next numbers in sequence ($3$, $4$, $5$, etc.), writing down successful pairs.
4. Stop searching when you reach a number that has already been listed as a partner in a previous pair.

This method guarantees you find every pair without creating duplicates.

Visual Worked Examples

A helpful way to visualize these combinations is by drawing arrays. An array arranges objects into equal rows and columns. The number of rows and the number of columns form a factor pair.

Example 1: Finding pairs for an even number

Question: Find all the factor pairs for $20$.

Method:

1. Start with $1$. We have the pair $1$ and $20$.
2. Try $2$. Because $20$ is even, $2 \times 10 = 20$.
3. Try $3$. The number $20$ cannot be divided evenly by $3$.
4. Try $4$. Since $4 \times 5 = 20$, this is a valid pair.
5. Try $5$. We have already listed $5$ with $4$, so we stop checking.

Answer: The factor pairs of $20$ are $(1, 20)$, $(2, 10)$, and $(4, 5)$.

Check: Multiply the pairs: $1 \times 20 = 20$, $2 \times 10 = 20$, and $4 \times 5 = 20$.

Example 2: Finding pairs for a square number

Question: List all the factor pairs for $36$.

Method:

1. Try $1$: $(1, 36)$.
2. Try $2$: $36$ is even, so $(2, 18)$.
3. Try $3$: $3 \times 12 = 36$, so $(3, 12)$.
4. Try $4$: $4 \times 9 = 36$, so $(4, 9)$.
5. Try $5$: $36$ does not end in $0$ or $5$, so $5$ is not a factor.
6. Try $6$: $6 \times 6 = 36$. The number multiplies by itself, meaning we have reached the square root and can stop.

Answer: The factor pairs are $(1, 36)$, $(2, 18)$, $(3, 12)$, $(4, 9)$, and $(6, 6)$.

Check: Every combination multiplies to $36$. Because we reached $6 \times 6$, we know no other pairs exist.

Example 3: Finding pairs for an odd number

Question: Find all the factor pairs for $45$.

Method:

1. Try $1$: $(1, 45)$.
2. Try $2$: $45$ is odd, so $2$ is not a factor.
3. Try $3$: $45$ divided by $3$ is $15$, so $(3, 15)$.
4. Try $4$: $45$ cannot be halved evenly twice, so $4$ is skipped.
5. Try $5$: $45$ ends in $5$, so $5 \times 9 = 45$. The pair is $(5, 9)$.
6. Try $6$, $7$, and $8$: None of these divide evenly into $45$.
7. Try $9$: We already listed $9$ with $5$, so we stop.

Answer: The factor pairs of $45$ are $(1, 45)$, $(3, 15)$, and $(5, 9)$.

Check: $1 \times 45 = 45$, $3 \times 15 = 45$, and $5 \times 9 = 45$.

How to Check the Answer

To verify your list of pairs, simply multiply each combination together. The product of every single pair must exactly equal your target number. If any pair results in a different total, you must remove or recalculate it.

Additionally, verify that you searched until the numbers began to repeat or you reached the square root of the target. This ensures you did not skip any valid multiples along the way.

Common Mistakes

A very frequent mistake is failing to include the number $1$ and the target number itself. For instance, stating that the only pairs for $12$ are $(2, 6)$ and $(3, 4)$ is incorrect because it misses $(1, 12)$.

Another common error is listing the same pair twice. In mathematics, a factor pair order does not matter. The pair $(3, 5)$ is identical to $(5, 3)$. You only write it once.

Finally, avoid forcing numbers that do not divide evenly. If there is a remainder, the numbers do not form a factor pair.

If you have leftover pieces when making an array, the numbers do not form a factor pair.