Which of the following numbers is not a perfect square ?

Options

A. 3600
B. 6400
C. 81000
D. 2500

Answer

To determine which of the given numbers is not a perfect square, we will analyze each number one by one using prime factorization and pairing method.

### Step 1: Analyze 3600
1. **Prime Factorization of 3600**:
- Start dividing by the smallest prime number (2):
- 3600 ÷ 2 = 1800
- 1800 ÷ 2 = 900
- 900 ÷ 2 = 450
- 450 ÷ 2 = 225
- Now, 225 is not divisible by 2, so we try the next prime number (3):
- 225 ÷ 3 = 75
- 75 ÷ 3 = 25
- Finally, 25 is divisible by 5:
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1
- The prime factorization of 3600 is:
- \( 3600 = 2^4 \times 3^2 \times 5^2 \)

2. **Checking for Perfect Square**:
- All prime factors have even powers (4, 2, and 2), hence 3600 is a perfect square.

### Step 2: Analyze 6400
1. **Prime Factorization of 6400**:
- Start dividing by 2:
- 6400 ÷ 2 = 3200
- 3200 ÷ 2 = 1600
- 1600 ÷ 2 = 800
- 800 ÷ 2 = 400
- 400 ÷ 2 = 200
- 200 ÷ 2 = 100
- 100 ÷ 2 = 50
- 50 ÷ 2 = 25
- Now, 25 is divisible by 5:
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1
- The prime factorization of 6400 is:
- \( 6400 = 2^7 \times 5^2 \)

2. **Checking for Perfect Square**:
- The power of 2 is odd (7), while the power of 5 is even (2). Hence, 6400 is **not** a perfect square.

### Step 3: Analyze 2704
1. **Prime Factorization of 2704**:
- Start dividing by 2:
- 2704 ÷ 2 = 1352
- 1352 ÷ 2 = 676
- 676 ÷ 2 = 338
- 338 ÷ 2 = 169
- Now, 169 is \( 13^2 \):
- 169 ÷ 13 = 13
- 13 ÷ 13 = 1
- The prime factorization of 2704 is:
- \( 2704 = 2^4 \times 13^2 \)

2. **Checking for Perfect Square**:
- All prime factors have even powers (4 and 2), hence 2704 is a perfect square.

### Step 4: Analyze 3969
1. **Prime Factorization of 3969**:
- Start dividing by 3:
- 3969 ÷ 3 = 1323
- 1323 ÷ 3 = 441
- 441 ÷ 3 = 147
- 147 ÷ 3 = 49
- Now, 49 is \( 7^2 \):
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1
- The prime factorization of 3969 is:
- \( 3969 = 3^4 \times 7^2 \)

2. **Checking for Perfect Square**:
- All prime factors have even powers (4 and 2), hence 3969 is a perfect square.

### Conclusion
From the analysis, we find that:
- 3600 is a perfect square.
- 6400 is **not** a perfect square.
- 2704 is a perfect square.
- 3969 is a perfect square.

Therefore, the number that is **not a perfect square** is **6400**.