Question

How many factors of \( 2^5 \times 3^6 \times 5^2 \) are perfect squares?

• A. 16
• B. 24
• C. 12
• D. 22

Hint:

To find the number of perfect square factors of a number, examine the exponents of its prime factorization. The exponents must be even for the number to be a perfect square. Count the number of even choices for each exponent and multiply them together.

Answer 1

The Correct Option is B

We are asked to find how many factors of the number \( N = 2^5 \times 3^6 \times 5^2 \) are perfect squares.
Step 1: Understand the condition for a number to be a perfect square. A number is a perfect square if all the exponents in its prime factorization are even.
Step 2: Express the general form of the factors of \( N \).
The factors of \( N \) will be of the form:
\[ 2^a \times 3^b \times 5^c \] where \( 0 \leq a \leq 5 \), \( 0 \leq b \leq 6 \), and \( 0 \leq c \leq 2 \). Step 3: Determine the conditions for the factors to be perfect squares. For the factor \( 2^a \times 3^b \times 5^c \) to be a perfect square, the exponents \( a \), \( b \), and \( c \) must all be even. - For \( a \), since \( a \) can range from 0 to 5, the possible even values of \( a \) are \( a = 0, 2, 4 \), so there are 3 choices for \( a \). - For \( b \), since \( b \) can range from 0 to 6, the possible even values of \( b \) are \( b = 0, 2, 4, 6 \), so there are 4 choices for \( b \). - For \( c \), since \( c \) can range from 0 to 2, the possible even values of \( c \) are \( c = 0, 2 \), so there are 2 choices for \( c \). Step 4: Calculate the total number of perfect square factors. The total number of perfect square factors is the product of the number of choices for \( a \), \( b \), and \( c \): \[ 3 \times 4 \times 2 = 24 \] Thus, there are 24 perfect square factors of \( N \). Therefore, the correct answer is option (B)