Is it possible for the square of a number to end with `5` zeroes ? Give reason.

Answer

To determine whether the square of a number can end with 5 zeros, we can follow these steps:

### Step 1: Understanding the Problem
We need to find out if a number can be squared to produce a result that ends with 5 zeros.

**Hint:** Think about what it means for a number to end with zeros.

### Step 2: Analyzing the Number of Zeros
When a number ends with zeros, it means that it is divisible by 10. Each zero at the end of a number corresponds to a factor of 10. Since \(10 = 2 \times 5\), each zero requires one factor of 2 and one factor of 5.

**Hint:** Consider how many factors of 2 and 5 are needed for a number to end with a certain number of zeros.

### Step 3: Squaring the Number
If a number \(n\) ends with \(k\) zeros, then it can be expressed as:
\[ n = m \times 10^k \]
where \(m\) is some integer. When we square \(n\), we have:
\[ n^2 = (m \times 10^k)^2 = m^2 \times 10^{2k} \]

**Hint:** What happens to the number of zeros when you square a number?

### Step 4: Determining the Number of Zeros in the Square
From the equation \(n^2 = m^2 \times 10^{2k}\), we see that the square of \(n\) will end with \(2k\) zeros. This means that the number of zeros at the end of the square is always even.

**Hint:** Think about the implications of having an even number of zeros.

### Step 5: Conclusion
Since 5 is an odd number, it is impossible for the square of any integer to end with 5 zeros. The number of zeros at the end of a square must always be even.

**Final Answer:** Therefore, it is not possible for the square of a number to end with 5 zeros.