Give reason to show that none of the number given below is a perfect square:
5372
5963
8457
9468
360
64000
2500000

Answer

To determine whether the numbers given are perfect squares, we can analyze their unit digits and the number of trailing zeros. Here’s a step-by-step solution:

### Step 1: Analyze the Unit Digits
The unit digit of a perfect square can only be 0, 1, 4, 5, 6, or 9. If a number has a unit digit of 2, 3, 7, or 8, it cannot be a perfect square.

- **5372**: The unit digit is **2**. Therefore, it cannot be a perfect square.
- **5963**: The unit digit is **3**. Therefore, it cannot be a perfect square.
- **8457**: The unit digit is **7**. Therefore, it cannot be a perfect square.
- **9468**: The unit digit is **8**. Therefore, it cannot be a perfect square.

### Step 2: Analyze the Trailing Zeros
A perfect square must have an even number of trailing zeros. If a number has an odd number of trailing zeros, it cannot be a perfect square.

- **360**: This number has **1 trailing zero** (360 = 36 × 10). Since 1 is odd, it cannot be a perfect square.
- **64000**: This number has **3 trailing zeros** (64000 = 64 × 1000). Since 3 is odd, it cannot be a perfect square.
- **2500000**: This number has **6 trailing zeros** (2500000 = 2500 × 1000). Since 6 is even, we need to check the other conditions. The unit digit is **0**, which is acceptable for a perfect square, but we also need to check the square root. The square root of 2500000 is 1581.1, which is not an integer. Therefore, it cannot be a perfect square.

### Conclusion
None of the numbers given (5372, 5963, 8457, 9468, 360, 64000, 2500000) are perfect squares.