How Can You Quickly Find the Factors of a Number?

Factors are the numbers that divide exactly into another number without leaving a remainder. A factor can never be greater than the number itself.

To find factors quickly, begin with the number 1 and ask:

Can this number divide evenly into the target number?

Since 1 divides into every integer, you can always start your factor list with 1 and the number itself.

Then proceed:

  • Increment your divisor by 1
  • For each new number, ask if it divides into the original number without leaving a remainder
  • If yes, add both the divisor and its corresponding quotient to your list
  • Stop once your divisor exceeds the square root of the original number

A number with only two factors — 1 and itself — is called a prime number. If it has more than two factors, it’s a composite number.

Step-by-Step: Find Factors of 10

Here’s how to find all the factors of 10:

  1. Start with the pair:
    $$ \text{Factors} = {1, 10} $$

  2. Check:
    $$ 2 \mid 10 \quad \text{(Yes)} \quad \Rightarrow \text{Add } 2 \text{ and } \frac{10}{2} = 5 $$
    $$ \text{Factors} = {1, 2, 5, 10} $$

  3. Check:
    $$ 3 \nmid 10 \quad \text{(No)} $$

  4. Since ( \sqrt{10} \approx 3.16 ), we stop here.

Final Result

  • All factors of 10:
    $$ {1, 2, 5, 10} $$

  • Since there are more than two factors, 10 is a composite number.

  • The highest proper factor (i.e. less than 10) is 5.

Summary

To find all the factors of a number:

  1. Start from 1 and go up to its square root
  2. If a number divides evenly, include it and its pair
  3. Stop once you’ve checked up to the square root

The factors of 10 are: $$ 1, 2, 5, 10 $$

Since 10 has more than two factors, it is a composite number.

Try your hand at finding the factors of 100 using this same approach!