What is the fastest way to find the highest common factor (HCF) between two numbers — and how can you be sure your answer is correct?

The quickest method is to:

  1. Identify all the factors of the smaller number.
  2. Check these from largest to smallest to see which one divides evenly into the larger number.

Once you find a factor that divides both numbers evenly, you’ve found the HCF.

Let’s walk through this with an example.

HCF of 6 and 15

Step-by-step:

  • The smaller number is 6.

  • Factors of 6: $[1, 2, 3, 6]$

  • Start from the largest:

    • $15 \div 6 = 2.5$ → ❌
    • $15 \div 3 = 5$ → ✅

So, the highest common factor of 6 and 15 is:

$$ \boxed{3} $$

How To Check Your Answer

You can double-check by doing the reverse:

  1. Find all factors of the larger number (15).
  2. Eliminate any that are larger than the smaller number (6).
  3. Test each factor in descending order to see which divides the smaller number.

  • Factors of 15: $[1, 3, 5, 15]$

  • Filter out values > 6 → $[1, 3, 5]$

    • $6 \div 5 = 1.2$ → ❌
    • $6 \div 3 = 2$ → ✅

So again, the HCF is 3.

Both methods confirm the same result.

Summary

The highest common factor (HCF) of 6 and 15 is:

$$ \boxed{3} $$

This is the largest number that divides evenly into both 6 and 15.
Knowing how to find the HCF is essential when simplifying fractions and solving arithmetic problems efficiently.