Question

What is the LCM of three positive integers \(x, y\) and \(z\)? Statements: (I) \(x\) and \(y\) are divisors of \(z\). (II) \(x+y+z=128\)

• A. Statement (I) alone is sufficient to answer the question, but (II) alone is not sufficient.
• B. Statement (II) alone is sufficient to answer the question, but (I) alone is not sufficient.
• C. Both the statements (I) and (II) are sufficient to answer the question, but neither statement alone is sufficient.
• D. Both the statements (I) and (II) together are not sufficient to answer the question and additional data are required.

Hint:

If all numbers divide one particular number, that number itself becomes their LCM.

Answer 1

The Correct Option is A

Concept: The Least Common Multiple (LCM) of numbers is the smallest number divisible by all of them. Important property: If one number is divisible by all the others, then that number itself is the LCM.

Step 1: Checking Statement (I).
Given: \[ x \text{ and } y \text{ are divisors of } z \] This means: \[ x \mid z \quad \text{and} \quad y \mid z \] So \(z\) is divisible by both \(x\) and \(y\). Hence, \(z\) itself is the smallest common multiple of \(x,y,z\). Thus: \[ LCM(x,y,z)=z \] So Statement (I) alone is sufficient.

Step 2: Checking Statement (II).
Given: \[ x+y+z=128 \] Many combinations are possible. Example: \[ (20,30,78) \] or \[ (10,18,100) \] Different combinations give different LCMs. So Statement (II) alone is not sufficient. Hence, Statement (I) alone is sufficient.