2023 · Q96
Question: What is the sum of two natural numbers?
Statement ILCM of the two numbers is 144.
Statement IIOne of the numbers is 72. Mark option (a) if the question can be answered by using one of the statements alone, but cannot be answered using the other statement alone. Mark option (b) if the question can be answered by using either statement alone. Mark option (c) if the question can be answered by using both the statements together, but cannot be answered using either statement alone. Mark option (d) if the question cannot be answered even by using both the statements together.
Elementary MathematicsHCF and LCM
- AThe question can be answered by using one of the statements alone, but cannot be answered using the other statement alone
- BThe question can be answered by using either statement alone
- CThe question can be answered by using both the statements together, but cannot be answered using either statement alone
- DThe question cannot be answered even by using both the statements together
Explanation
Write the prime factorization: 72 = 2³·3² and 144 = 2⁴·3². If lcm(72, b) = 144, then b = 2ᵃ·3ʳ with a ≤ 4, j ≤ 2, max(a,3) = 4 (so a = 4) and max(j,2) = 2 (so j ≤ 2). Hence b ∈ {16, 48, 144} — three possibilities, so the sum is not uniquely determined even with both statements. The answer is (D): the question cannot be answered even using both statements together.