Question

If X, Y and Z are three numbers, such that LCM of X and Y is X and LCM of Y and Z is Y, then find the HCF of X, Y and Z.

• A. \(X \)
• B. \(Y \)
• C. \(Z \)
• D. \(XY \)
• E. \(YZ \)

Hint:

If $\text{LCM}(a,b)=a$, then $a$ is divisible by $b$. Use divisibility chains to quickly find HCF.

Answer 1

The Correct Option is B

Concept:
• If $\text{LCM}(a, b) = a$, then $a$ is a multiple of $b$ (i.e., $b \mid a$)
• HCF is the greatest common divisor among the numbers
Step 1: Interpret given conditions.
Given: \[ \text{LCM}(X, Y) = X \Rightarrow X \text{ is a multiple of } Y \Rightarrow Y \mid X \] \[ \text{LCM}(Y, Z) = Y \Rightarrow Y \text{ is a multiple of } Z \Rightarrow Z \mid Y \]

Step 2: Understand number relationship.
From above: \[ Z \mid Y \mid X \] So the numbers are in the form: \[ X \ge Y \ge Z \] and each divides the next.

Step 3: Find HCF.
Since $Z \mid Y$ and $Y \mid X$, the greatest common factor among $X, Y, Z$ is: \[ \text{HCF}(X, Y, Z) = Y \]