Question: A positive integer x is divisible by 2,3, and 5. What is the smallest three-digit number that is divisible by 2, 3, 5, and 3x?

1. 120
2. 150
3. 180
4. 210
5. 300

Correct Answer: (C)

Solution and Explanation:
Approach Solution 1:

The problem statement states that a positive integer x is divisible by 2,3, and 5. It is required to find out the smallest three-digit number that is divisible by 2, 3, 5, and 3x.

To solve this problem, we need to find the least common factor of 2, 3 and 5.
LCM of 2,3 and 5 = 30
Hence we can say, the value of x = 30y, where y is an integer.
Therefore, the least three-digit no. that is divisible by 2,3,5 and 90 = LCM of 2,3,5 and 90 = 90

Hence, the no. must be a multiple of 90.
Therefore, after analysing the options, we can infer that 180 is the least multiple.

Approach Solution 2:

The problem statement states that:

Given:

• a positive integer x is divisible by 2,3, and 5.

Find out:

• The smallest three-digit number that is divisible by 2, 3, 5, and 3x

Since it is given that x is divisible by 2, 3, 5
Therefore, we can write, x is a multiple of 2*3*5, i.e x= 30
Hence, the value of x may be 30, 60 ... so on.
The smallest three-digit number divisible by 2, 3, 5, and 3x can be said to be divisible by LCM(2, 3, 5, 3* 30 or 60) = Multiple of 90 = 180

Therefore, the smallest three-digit number is 180.

Approach Solution 3:

The problem statement says that a positive integer x is divisible by 2,3, and 5.

It is asked to find the smallest three-digit number divisible by 2, 3, 5, and 3x

Given, x is divisible by 2,3, and 5.
Therefore, x is a multiple of 2,3, & 5
Hence, x = LCM (2,3, & 5) = 2*3*5*d = 30d, where d = 0(1)n, n is a positive integer.
Therefore, the smallest three-digit number divisible by 2, 3, 5, and 3x = LCM (2,3,5, & 3x) = LCM (2,3,5, & 3*30d) = 3*30d = 90d.
Hence, the smallest three-digit number divisible by 2, 3, 5, and 3x = 90, 180, or 270, ....

As per the options given, the smallest three-digit number is 180.

“A positive integer x is divisible by 2,3, and 5. What is the smallest”- is a topic of the GMAT Quantitative reasoning section of the GMAT exam. This topic has been taken from the book “GMAT Official Guide 2021”. The GMAT Problem Solving questions test the candidates’ abilities in calculating the numerical problems. GMAT Quant practice papers help the candidates to analyse various questions that will enable them to improve their mathematical understanding.

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