Sum of digits of the smallest number by which 1440 be multiplied so that it becomes a perfect cube, is

Options

A. 4
B. 6
C. 7
D. 8

Answer

To find the smallest number by which 1440 must be multiplied so that it becomes a perfect cube, we will follow these steps:

### Step 1: Prime Factorization of 1440
We start by finding the prime factorization of 1440.

1. Divide by 2:
- \( 1440 \div 2 = 720 \)
2. Divide by 2:
- \( 720 \div 2 = 360 \)
3. Divide by 2:
- \( 360 \div 2 = 180 \)
4. Divide by 2:
- \( 180 \div 2 = 90 \)
5. Divide by 2:
- \( 90 \div 2 = 45 \)
6. Divide by 3:
- \( 45 \div 3 = 15 \)
7. Divide by 3:
- \( 15 \div 3 = 5 \)
8. Divide by 5:
- \( 5 \div 5 = 1 \)

So, the prime factorization of 1440 is:
\[ 1440 = 2^5 \times 3^2 \times 5^1 \]

### Step 2: Determine the Exponents for Perfect Cube
For a number to be a perfect cube, all the exponents in its prime factorization must be multiples of 3.

- The exponent of \( 2 \) is \( 5 \). To make it a multiple of 3, we need \( 6 \) (the next multiple of 3). Thus, we need \( 6 - 5 = 1 \) more \( 2 \).
- The exponent of \( 3 \) is \( 2 \). To make it a multiple of 3, we need \( 3 \). Thus, we need \( 3 - 2 = 1 \) more \( 3 \).
- The exponent of \( 5 \) is \( 1 \). To make it a multiple of 3, we need \( 3 \). Thus, we need \( 3 - 1 = 2 \) more \( 5 \).

### Step 3: Calculate the Smallest Number to Multiply
Now we can find the smallest number we need to multiply by:
\[ 2^1 \times 3^1 \times 5^2 = 2 \times 3 \times 25 = 150 \]

### Step 4: Sum of the Digits of the Result
Now, we calculate the sum of the digits of \( 150 \):
- The digits are \( 1, 5, 0 \).
- The sum is \( 1 + 5 + 0 = 6 \).

### Final Answer
The sum of the digits of the smallest number by which 1440 must be multiplied to become a perfect cube is \( 6 \).

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