The smallest value by which 63520 is substracted to make it a perfect square?

Options

A. 16
B. 20
C. 24
D. 30

Answer

To find the smallest value by which 63520 must be subtracted to make it a perfect square, we can follow these steps:

### Step 1: Prime Factorization of 63520
We start by performing the prime factorization of 63520.

1. Divide by 2:
- 63520 ÷ 2 = 31760
- 31760 ÷ 2 = 15880
- 15880 ÷ 2 = 7940
- 7940 ÷ 2 = 3970
- 3970 ÷ 2 = 1985 (stop here as 1985 is odd)

2. Now, we factor 1985:
- 1985 ÷ 5 = 397 (since 1985 ends with 5)
- 397 is a prime number.

So, the prime factorization of 63520 is:
\[ 63520 = 2^5 \times 5^1 \times 397^1 \]

### Step 2: Determine the Exponents
To form a perfect square, all the exponents in the prime factorization must be even.

- The exponent of 2 is 5 (which is odd).
- The exponent of 5 is 1 (which is odd).
- The exponent of 397 is 1 (which is odd).

### Step 3: Make Exponents Even
To make the exponents even, we need to adjust them:

1. For \(2^5\), we need to reduce it to \(2^4\) (subtract 1).
2. For \(5^1\), we need to reduce it to \(5^0\) (subtract 1).
3. For \(397^1\), we need to reduce it to \(397^0\) (subtract 1).

### Step 4: Calculate the Perfect Square
Now, we calculate the smallest perfect square that can be formed by adjusting these factors:

1. If we reduce \(2^5\) to \(2^4\), we subtract \(16\) (since \(2^4 = 16\)).
2. If we reduce \(5^1\) to \(5^0\), we subtract \(5\) (since \(5^0 = 1\)).
3. If we reduce \(397^1\) to \(397^0\), we subtract \(397\) (since \(397^0 = 1\)).

### Step 5: Calculate the Total Subtraction
To find the total value we need to subtract from 63520 to make it a perfect square, we need to find the smallest number that will make the remaining number a perfect square:

- The smallest adjustment needed is to subtract \(16\) from \(63520\).

### Conclusion
Thus, the smallest value by which 63520 must be subtracted to make it a perfect square is **16**.

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