56 64 simplified

2 min read 10-03-2025

The fraction 56/64 might look daunting at first, but simplifying it is easier than you think! This guide will walk you through the process step-by-step, explaining the concepts involved and providing a clear solution. We'll also cover finding the greatest common divisor (GCD), a crucial step in simplifying fractions.

Understanding Fraction Simplification

Simplifying a fraction means reducing it to its lowest terms. This means finding an equivalent fraction where the numerator (top number) and the denominator (bottom number) have no common factors other than 1. This makes the fraction easier to understand and work with.

Finding the Greatest Common Divisor (GCD)

The key to simplifying 56/64 is finding the greatest common divisor (GCD) of 56 and 64. The GCD is the largest number that divides evenly into both numbers. There are several ways to find the GCD:

1. Listing Factors

List all the factors of 56 and 64:

Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
Factors of 64: 1, 2, 4, 8, 16, 32, 64

The largest number that appears in both lists is 8. Therefore, the GCD of 56 and 64 is 8.

2. Prime Factorization

This method involves breaking down each number into its prime factors:

56 = 2 x 2 x 2 x 7 = 2³ x 7
64 = 2 x 2 x 2 x 2 x 2 x 2 = 2⁶

The common prime factors are three 2's (2³). Therefore, the GCD is 2 x 2 x 2 = 8.

3. Euclidean Algorithm (for larger numbers)

The Euclidean algorithm is a more efficient method for finding the GCD of larger numbers. It involves repeatedly applying the division algorithm until the remainder is 0. We won't delve into this here, as it's not necessary for smaller numbers like 56 and 64.

Simplifying 56/64

Now that we know the GCD of 56 and 64 is 8, we can simplify the fraction:

Divide both the numerator and the denominator by the GCD (8):

56 ÷ 8 = 7 64 ÷ 8 = 8

Therefore, the simplified fraction is 7/8.

Verifying the Simplification

To ensure our simplification is correct, we can check if 7 and 8 share any common factors other than 1. They don't, confirming that 7/8 is the simplest form of 56/64.

Conclusion: Mastering Fraction Simplification

Simplifying fractions like 56/64 is a fundamental skill in mathematics. By understanding the concept of the greatest common divisor and applying the appropriate method, you can easily reduce fractions to their lowest terms. Remember, the key is to divide both the numerator and the denominator by their GCD. Now you can confidently tackle other fraction simplification problems!