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Math  /  Numbers & Operations

QuestionFind the Least Common Multiple (LCM) a) 72 and 132

Studdy Solution

STEP 1

1. The Least Common Multiple (LCM) of two integers is the smallest positive integer that is divisible by both numbers.
2. To find the LCM, we can use the prime factorization of each number.
3. We utilize the highest power of each prime factor that appears in the factorizations.
4. The LCM can also be found using the relation LCM(a,b)=∣a⋅b∣GCD(a,b) \text{LCM}(a, b) = \frac{|a \cdot b|}{\text{GCD}(a, b)} , where GCD is the Greatest Common Divisor.

STEP 2

1. Find the prime factorization of 72.
2. Find the prime factorization of 132.
3. Determine the highest power of each prime factor present in the factorizations.
4. Multiply these highest powers to obtain the LCM.

STEP 3

Find the prime factorization of 72.
72=23×32 72 = 2^3 \times 3^2

STEP 4

Find the prime factorization of 132.
132=22×3×11 132 = 2^2 \times 3 \times 11

STEP 5

Determine the highest power of each prime factor appearing in the factorizations.
For 2 2 : The highest power is 23 2^3 .
For 3 3 : The highest power is 32 3^2 .
For 11 11 : The highest power is 111 11^1 .

STEP 6

Multiply these highest powers to obtain the LCM.
LCM(72,132)=23×32×111=8×9×11=792 \text{LCM}(72, 132) = 2^3 \times 3^2 \times 11^1 = 8 \times 9 \times 11 = 792
Solution: The Least Common Multiple (LCM) of 72 and 132 is 792.

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