What is the Smallest Positive Integer k such that \(126 * \sqrt k\) GMAT Problem Solving

Question: What is the smallest positive integer k such that \(126 * \sqrt k\) is the square of a positive integer?

1. 14
2. 36
3. 144
4. 196
5. 441

Correct Answer: A
Solution and Explanation:
Approach Solution 1:

126 can be broken down into its factors as written below:

\(126 = 2 *3^2*7,\)

So in order \(126 * \sqrt k\) to be a square of an integer, \(\sqrt k\) must have the powers of 2 and 7 to be an even number.

Then only the least value of \(\sqrt k\) must be equal to 2*7 = 14

This makes the least value of k to be equal to 14^2 = 196

Approach Solution 2:

Factorization is a simple method for answering this query.

\(126 * \sqrt k\)

\(= 2*3*3*7*\sqrt k\)

=> (3^2)*14*\(\sqrt k\)

To make it a square of a positive number, we need an additional 14, therefore the answer is \(\sqrt k\) = 14
K=196
D is the correct answer.

“What is the smallest positive integer k such that" - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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