Question

if y is a positive integer, for how many different values of y is \\(\sqrt3{\frac{144}{y}}\\) a whole number?

1
2
6
15

Explanation:

Set up the condition for a whole number

Using the Integer Divisibility knowledge point

$$
LATEXBLOCK0
$$

Analyze constraints on the integer variables

Since \(y\) must be a positive integer, we have:

$$
LATEXBLOCK1
$$

Find perfect cube divisors of 144

We find all positive integers \(k\) such that \(k^3\) divides \(144\).
First, find the prime factorization of \(144\):

$$ 144 = 12^2 = (2^2 \cdot 3)^2 = 2^4 \cdot 3^2 $$

For \(k^3\) to divide \(2^4 \cdot 3^2\), the prime factorization of \(k^3\) can only contain primes \(2\) and \(3\) with exponents less than or equal to those in \(144\):

$$
LATEXBLOCK2
$$

Count the valid values of y

There are exactly \(2\) positive integer values of \(y\) (\(144\) and \(18\)) that satisfy the condition.

Answer:

• (A) 1
• (B) 2 (Correct answer)
• (C) 6
• (D) 15