Question

How many factors of \(2^4\times3^5\times10^4\) are perfect squares which are greater than 1 ?

• A. 44
• B. 38
• C. 45
• D. 22

Answer 1

The Correct Option is A

We are given the number: 

\[ 2^4 \times 3^5 \times 10^4 = 2^4 \times 3^5 \times (2 \times 5)^4 = 2^4 \times 3^5 \times 2^4 \times 5^4 = 2^8 \times 3^5 \times 5^4 \]

Now, to count the number of perfect square factors, we consider only the even powers of each prime factor.

Step 1: Choose even powers for each prime factor

• For \( 2^8 \), possible even powers are: \( 2^0, 2^2, 2^4, 2^6, 2^8 \) ⇒ 5 options
• For \( 3^5 \), even powers are: \( 3^0, 3^2, 3^4 \) ⇒ 3 options
• For \( 5^4 \), even powers are: \( 5^0, 5^2, 5^4 \) ⇒ 3 options

Step 2: Multiply the options to get total perfect square factors

\[ \text{Total perfect square factors} = 5 \times 3 \times 3 = 45 \]

Step 3: Exclude the number 1

Since 1 is also included as a perfect square factor, we subtract it:

\[ \text{Required count (excluding 1)} = 45 - 1 = \boxed{44} \]