The least possible value of A for which `90 xx A` is a perfect cube is

Options

A. a) 200
B. b) 300
C. c) 500
D. d) 600

Answer

To find the least possible value of \( A \) for which \( 90 \times A \) is a perfect cube, we can follow these steps:

### Step 1: Factorize 90
First, we need to factorize the number 90 into its prime factors.
\[ 90 = 9 \times 10 = 3^2 \times (2 \times 5) = 2^1 \times 3^2 \times 5^1 \]

### Step 2: Understand the condition for a perfect cube
A number is a perfect cube if all the exponents in its prime factorization are multiples of 3.

### Step 3: Analyze the exponents of the prime factors in 90
From the factorization \( 90 = 2^1 \times 3^2 \times 5^1 \):
- The exponent of 2 is 1.
- The exponent of 3 is 2.
- The exponent of 5 is 1.

### Step 4: Determine how much more is needed for each prime factor
To make each exponent a multiple of 3:
- For \( 2^1 \): We need \( 3 - 1 = 2 \) more factors of 2.
- For \( 3^2 \): We need \( 3 - 2 = 1 \) more factor of 3.
- For \( 5^1 \): We need \( 3 - 1 = 2 \) more factors of 5.

### Step 5: Calculate the value of \( A \)
Now we can express \( A \) in terms of the necessary prime factors:
\[ A = 2^2 \times 3^1 \times 5^2 \]
Calculating this:
\[ A = 4 \times 3 \times 25 \]
Calculating step-by-step:
1. \( 4 \times 3 = 12 \)
2. \( 12 \times 25 = 300 \)

### Conclusion
Thus, the least possible value of \( A \) for which \( 90 \times A \) is a perfect cube is:
\[ \boxed{300} \]