How Can You Quickly Find the Factors of a Number?

Factors are the numbers that divide exactly into another number, leaving no remainder. A factor cannot be greater than the number it divides.

To find factors quickly:

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

If a number has only two factors (1 and itself), it is a prime number. Otherwise, it is a composite number.

Step-by-Step: Find the Factors of 144

Let’s work through this together.

Start with: $$ {1, 144} $$

  • Check: $$ 2 \mid 144 \Rightarrow {1, 2, 72, 144} $$

  • Check: $$ 3 \mid 144 \Rightarrow {1, 2, 3, 48, 72, 144} $$

  • Check: $$ 4 \mid 144 \Rightarrow {1, 2, 3, 4, 36, 48, 72, 144} $$

  • $5 \nmid 144$

  • Check: $$ 6 \mid 144 \Rightarrow {1, 2, 3, 4, 6, 24, 36, 48, 72, 144} $$

  • $7 \nmid 144$

  • Check: $$ 8 \mid 144 \Rightarrow {1, 2, 3, 4, 6, 8, 18, 24, 36, 48, 72, 144} $$

  • Check: $$ 9 \mid 144 \Rightarrow {1, 2, 3, 4, 6, 8, 9, 16, 18, 24, 36, 48, 72, 144} $$

  • $10 \nmid 144$, $11 \nmid 144$

  • Check: $$ 12 \mid 144 \Rightarrow \text{already in list via 12 × 12} $$

Since $\sqrt{144} = 12$, we stop here.

Final Result

  • Factors of 144: $$ {1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144} $$

  • Highest proper factor: 72

  • 144 is a composite number

Summary

The factors of 144 are: $$ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 $$

Since it has more than two factors, 144 is a composite number.