How to Simplify Fractions: Finding the Greatest Common Divisor

Method 1: Prime Factorization

Break both numbers into primes. 84 = 2 × 2 × 3 × 7 and 126 = 2 × 3 × 3 × 7. The common primes are 2, 3, and 7, so the GCD is 2 × 3 × 7 = 42. Divide top and bottom by 42 to get 2/3.

This method is intuitive and works well for small numbers, but it gets slow for large numbers that are hard to factor.

Method 2: The Euclidean Algorithm

The Euclidean algorithm finds the GCD without factoring. Divide the larger number by the smaller one, take the remainder, and repeat with the smaller number and the remainder until the remainder is zero. The last non-zero remainder is the GCD.

For 126 and 84: 126 = 1 × 84 + 42. Then 84 = 2 × 42 + 0. The remainder is zero, so the GCD is 42. For 1071 and 462: 1071 = 2 × 462 + 147, 462 = 3 × 147 + 21, 147 = 7 × 21 + 0, so the GCD is 21.

• Divide the larger number by the smaller number.
• Replace the larger number with the smaller and the smaller with the remainder.
• Repeat until the remainder is 0.
• The last non-zero remainder is the GCD.

Shortcuts and Checks

Quick divisibility tests help you spot common factors. A number is divisible by 2 if it ends in an even digit, by 5 if it ends in 0 or 5, by 3 if its digits sum to a multiple of 3, and by 9 if its digits sum to a multiple of 9.

If you do not find the GCD right away, divide by any common factor you spot and repeat. The result is the same, it just takes more steps.

Why Simplifying Matters

Reduced fractions are easier to compare, add, and interpret. 42/63 and 2/3 are equal, but only one of them is obviously two thirds. When adding fractions, using the least common denominator rather than just any common denominator also keeps numbers small.

A fraction calculator reduces its answers automatically, but knowing the method helps you check results and work without one.