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Problem 21

# Find the greatest common divisor and the least common multiple of 16 and 12 .

Expert verified
The greatest common divisor (GCD) of 16 and 12 is $$4$$, and their least common multiple (LCM) is $$48$$.
See the step by step solution

## Step 1: Prime Factorization

Find the prime factors of the given numbers (16 and 12).

## Step 2: 16's Prime Factors

Divide 16 by its smallest prime factor (2) until there's only 1 left:  16 ÷ 2 = 8 8 ÷ 2 = 4 4 ÷ 2 = 2 2 ÷ 2 = 1  The prime factorization of 16 is: 2 × 2 × 2 × 2

## Step 3: 12's Prime Factors

Divide 12 by its smallest prime factor (2) and then by other prime factors until there's only 1 left:  12 ÷ 2 = 6 6 ÷ 2 = 3 3 ÷ 3 = 1  The prime factorization of 12 is: 2 × 2 × 3

## Step 2: Finding the GCD

Find the common prime factors and their lowest power, then multiply to get the GCD: Common prime factors are 2. The lowest power of 2 found in both factorizations is 2 to the power of 2. So, the GCD is: 2 × 2 = $$4$$

## Step 3: Finding the LCM

Use the GCD and the formula, $$LCM(a, b) = |a * b| / GCD(a, b)$$, to find the LCM of 16 and 12: LCM(16, 12) = |16 × 12| / 4 LCM(16, 12) = 192 / 4 The LCM is: $$48$$ Thus, the greatest common divisor of 16 and 12 is 4, and their least common multiple is 48.

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