GCD, LCM and Fractions: How They Connect

Two ideas from school arithmetic make fractions easy: the greatest common divisor, which simplifies them, and the least common multiple, which lets you add them.

Greatest common divisor (GCD)

The GCD, also called the highest common factor, is the biggest number that divides two numbers exactly. Euclid's method finds it by repeated division. For 48 and 18: 48 = 2 x 18 + 12, then 18 = 1 x 12 + 6, then 12 = 2 x 6 + 0. The last divisor, 6, is the GCD.

Least common multiple (LCM)

The LCM is the smallest number that both numbers divide into. Multiply the two numbers and divide by the GCD: 48 x 18 / 6 = 144. The GCD & LCM Calculator does this for any list of whole numbers, of any size, and shows the prime factors and Euclid's steps.

Simplifying a fraction

Divide the top and bottom by their GCD. For 18/48 the GCD is 6, so the fraction is 3/8. It can't be reduced any further.

Adding fractions

Fractions can only be added when their bottoms match. Use the LCM of the denominators as the common bottom. For 1/4 + 1/6 the LCM of 4 and 6 is 12, so you add 3/12 and 2/12 to get 5/12. Subtracting works the same way. To multiply, multiply tops and bottoms, and to divide, flip the second fraction and multiply.

Mixed numbers and decimals

• A mixed number like 2 1/3 is the improper fraction 7/3.
• Every fraction is a decimal that ends or repeats. 1/4 is 0.25, while 1/3 is 0.333... with the 3 repeating.
• The Fraction Calculator does all four operations, simplifies the result, and shows it as a mixed number, a decimal and a percentage.
• For everything else, the Scientific Calculator handles powers, roots and brackets.