If LCM of 8, 12 and K is 144, then what is the value of K?
2021
If LCM of 8, 12 and K is 144, then what is the value of K?
Answer: B. 144 — Solution: Use prime factorizations and the definition of LCM. Prime factorizations: 8 = 2^3; 12 = 2^2 × 3; 144 = 2^4 × 3^2. LCM of 8 and 12 is 24 = 2^3 × 3.…
- A.
12
- B.
144
- C.
18
- D.
48
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Correct answer: B
Solution: Use prime factorizations and the definition of LCM.
Prime factorizations: 8 = 2^3; 12 = 2^2 × 3; 144 = 2^4 × 3^2.
LCM of 8 and 12 is 24 = 2^3 × 3.
We need K such that LCM(24, K) = 144. For each prime, the exponent in the LCM is the maximum of its exponents in the numbers being combined. So the exponent of 2 in K must be 4 (since max(3, exp2(K)) = 4) and the exponent of 3 in K must be 2 (since max(1, exp3(K)) = 2).
Thus K = 2^4 × 3^2 = 144.
Answer: 144
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