How Can You Quickly Find the Factors of a Number?

Factors are numbers that divide evenly into another number — that is, without leaving a remainder. A factor must always be less than or equal to the number it’s dividing.

To find factors quickly:

  1. Start with 1 and the original number.
  2. Check each number up to the square root of the original.
  3. If it divides evenly, record the divisor and its quotient.

If a number has only two factors — 1 and itself — it is prime. If it has more than two, it is composite.

Step-by-Step: Find the Factors of 108

Let’s work step by step:

  • Start with: $$ {1, 108} $$

  • Check: $$ 2 \mid 108 \Rightarrow \text{Add } 2, \frac{108}{2} = 54 $$ $$ {1, 2, 54, 108} $$

  • Check: $$ 3 \mid 108 \Rightarrow \text{Add } 3, \frac{108}{3} = 36 $$ $$ {1, 2, 3, 36, 54, 108} $$

  • Check: $$ 4 \mid 108 \Rightarrow \text{Add } 4, \frac{108}{4} = 27 $$ $$ {1, 2, 3, 4, 27, 36, 54, 108} $$

  • $5 \nmid 108$ (remainder = 3)

  • Check: $$ 6 \mid 108 \Rightarrow \text{Add } 6, \frac{108}{6} = 18 $$ $$ {1, 2, 3, 4, 6, 18, 27, 36, 54, 108} $$

  • $7 \nmid 108$, $8 \nmid 108$

  • Check: $$ 9 \mid 108 \Rightarrow \text{Add } 9, \frac{108}{9} = 12 $$ $$ {1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108} $$

  • $10 \nmid 108$

Since $\sqrt{108} \approx 10.4$, we can stop here.

Final Result

  • Factors of 108: $$ {1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108} $$

  • Highest proper factor: 54

  • 108 is a composite number.

Summary

The factors of a number are all the values that divide into it exactly.

Factors of 108: $$ 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108 $$

Because it has more than two factors, 108 is a composite number.