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Problem 22

# Find the greatest common divisor of 24 and 40 . Also, find the least common multiple of 24 and 40 .

Expert verified
The greatest common divisor (GCD) of 24 and 40 is 8, and the least common multiple (LCM) of 24 and 40 is 120.
See the step by step solution

## Step 1: Find the GCD using the Euclidean Algorithm

First, we'll use the Euclidean algorithm to find the greatest common divisor (GCD) of 24 and 40. The Euclidean algorithm is a way to find the GCD of two integers by repeatedly applying the formula: GCD(a, b) = GCD(b, a % b), where the % symbol represents the remainder function. a = 40 b = 24 1. GCD(40, 24) = GCD(24, 40 % 24) 2. GCD(24, 16) = GCD(16, 24 % 16) 3. GCD(16, 8) = GCD(8, 16 % 8) After the last step, we get GCD(16, 8) = GCD(8, 0), which means the GCD is 8.

## Step 2: Find the LCM using the formula LCM(a, b) = (a * b) / GCD(a, b)

To find the least common multiple (LCM) of 24 and 40, we can use the formula LCM(a, b) = (a * b) / GCD(a, b). From the previous step, we already know that the GCD of 24 and 40 is 8. LCM(24, 40) = (24 * 40) / 8 Calculate the numerator and the denominator: Numerator = 24 * 40 = 960 Denominator = 8 Now, divide the numerator by the denominator: LCM(24, 40) = 960 / 8 = 120 The least common multiple of 24 and 40 is 120. In summary, the greatest common divisor of 24 and 40 is 8, and the least common multiple of 24 and 40 is 120.

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