Is `180` a perfect square ?

Answer

To determine if 180 is a perfect square, we can use the prime factorization method. Here’s a step-by-step solution:

### Step 1: Prime Factorization of 180
We start by dividing 180 by the smallest prime number, which is 2.

- **180 ÷ 2 = 90**

Next, we divide 90 by 2 again.

- **90 ÷ 2 = 45**

Now, we divide 45 by the next smallest prime number, which is 3.

- **45 ÷ 3 = 15**

We divide 15 by 3 again.

- **15 ÷ 3 = 5**

Finally, we divide 5 by 5.

- **5 ÷ 5 = 1**

### Step 2: Write the Prime Factorization
From the divisions, we can write the prime factorization of 180:

- **180 = 2 × 2 × 3 × 3 × 5**

This can also be expressed using exponents:

- **180 = 2² × 3² × 5¹**

### Step 3: Check for Perfect Squares
A number is a perfect square if all the prime factors in its factorization have even exponents.

In our factorization:

- The exponent of 2 is 2 (even)
- The exponent of 3 is 2 (even)
- The exponent of 5 is 1 (odd)

Since the exponent of 5 is odd, it means that 5 does not have a pair.

### Conclusion
Since there is at least one prime factor (5) that does not have an even exponent, we conclude that:

- **180 is not a perfect square.**

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