If LCM of 8, 12 and K is 144, then what is the value of K?

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