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

Options

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

Answer

To solve the problem of finding the sum of the digits of the smallest number by which 1440 should be multiplied to become a perfect cube, we can follow these steps:

### Step 1: Prime Factorization of 1440
First, we need to find the prime factorization of 1440.

1. Divide 1440 by 2 (the smallest prime number):
- 1440 ÷ 2 = 720
- 720 ÷ 2 = 360
- 360 ÷ 2 = 180
- 180 ÷ 2 = 90
- 90 ÷ 2 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 ÷ 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.

- For \(2^5\): The nearest multiple of 3 is 6. We need \(6 - 5 = 1\) more \(2\).
- For \(3^2\): The nearest multiple of 3 is 3. We need \(3 - 2 = 1\) more \(3\).
- For \(5^1\): The nearest multiple of 3 is 3. We need \(3 - 1 = 2\) more \(5\).

### Step 3: Calculate the Smallest Number to Multiply
To make 1440 a perfect cube, we need to multiply it by:
\[ 2^1 \times 3^1 \times 5^2 \]

Calculating this gives:
\[ 2^1 = 2 \]
\[ 3^1 = 3 \]
\[ 5^2 = 25 \]

Now, multiply these together:
\[ 2 \times 3 \times 25 = 6 \times 25 = 150 \]

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

### Final Answer
The sum of the digits of the smallest number by which 1440 should be multiplied to become a perfect cube is **6**.

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