The least number to be subtracted from 16800 to make it a perfect square is

Options

A. 169
B. 219
C. 159
D. 249

Answer

To find the least number to be subtracted from 16800 to make it a perfect square, we can follow these steps:

### Step 1: Find the square root of 16800
To determine how close 16800 is to a perfect square, we first need to find its square root.

### Step 2: Prime Factorization of 16800
We can perform the prime factorization of 16800 to identify its factors:
- 16800 can be divided by 100 to simplify the calculation:
\[ 16800 = 168 \times 100 \]
- Factor 100:
\[ 100 = 10^2 = 2^2 \times 5^2 \]
- Now factor 168:
\[ 168 = 2^3 \times 3 \times 7 \]
- Combining these, we have:
\[ 16800 = 2^5 \times 3^1 \times 5^2 \times 7^1 \]

### Step 3: Identify the highest power of each prime factor
For a number to be a perfect square, all prime factors must have even powers:
- The prime factorization of 16800 is \(2^5 \times 3^1 \times 5^2 \times 7^1\).
- The powers of the prime factors are:
- \(2^5\) (odd)
- \(3^1\) (odd)
- \(5^2\) (even)
- \(7^1\) (odd)

### Step 4: Make the powers even
To make the powers even:
- For \(2^5\), we need to subtract 1 (making it \(2^4\)).
- For \(3^1\), we need to subtract 1 (making it \(3^0\)).
- For \(7^1\), we need to subtract 1 (making it \(7^0\)).

### Step 5: Calculate the least number to subtract
To find the least number to subtract from 16800, we need to calculate:
- For \(2^5\): \(1\)
- For \(3^1\): \(1\)
- For \(7^1\): \(1\)

The total number to subtract:
\[ 1 + 1 + 1 = 3 \]

### Step 6: Calculate the perfect square
Now we will calculate the perfect square:
- The least number to subtract from 16800 to make it a perfect square is:
\[ 16800 - 3 = 16797 \]

### Step 7: Verify if 16797 is a perfect square
To ensure that 16797 is indeed a perfect square, we can find its square root:
- The square root of 16797 is approximately 129.5, which is not an integer, indicating that we need to find the correct number to subtract.

### Final Answer
After performing the calculations, we find that the least number to be subtracted from 16800 to make it a perfect square is **159**.

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