Here’s a question on LCM-GCD for you. Apply your understanding of Data Sufficiency to identify the right option choice. Find the solution below in case you have any doubt.

## Question :

If x is a positive integer, what is the least common multiple of x, 4x, and 15?

1. Least common multiple of x and 4x is 4.
2. Least common multiple of 3x and 15 is 15.

(A) Statement (1) ALONE is sufficient, but Statement (2) alone is not sufficient.

(B) Statement (2) ALONE is sufficient, but Statement (1) alone is not sufficient.

(C) Both statement TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

(D) Each statement ALONE is sufficient.

(E) Statement (1) and (2) TOGETHER are not sufficient.

## Solution :

#### Step 1: Analyse Question Stem

• x is a positive integer.

We need to find LCM(x, 4x, 15)

#### Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE

Statement 1: Least common multiple of x and 4x is 4

• LCM (x, 4x) = 4
• This means that x = 1
• Thus, the LCM of (x, 4x, 15) = LCM (1, 4, 15 )= 60

Note that we don’t even need to find the value of x, to find the LCM of (x, 4x, 15)

• If we know LCM of (x, 4x) = 4
• Then we just need to find the LCM of (4,15) = 60

Hence, statement 1 is sufficient, so we can eliminate options B, C, and E.

Statement 2: Least common multiple of 3x and 15 is 15

• LCM (3*x, 3*5) = 3*5
• x is either 1 or 5
• If x = 5,
• Then, LCM (5, 4*5, 3*5) = 60
• If x = 1,
• Then, LCM (1, 4*1, 3*5) = 60

Since in either case, the LCM is 60, Statement 2 is also sufficient. Hence, the correct answer is Option D.

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