The square of a natural mumber (other than 1) is either a multiple of 3 or exceeds a multiple of 3 by _____

Options

A. 3
B. 2
C. 4
D. 1

Answer

To solve the problem, we need to analyze the square of a natural number (other than 1) and determine its relationship with multiples of 3.

### Step-by-Step Solution:

1. **Understanding Natural Numbers**:
Natural numbers are the set of positive integers starting from 1, 2, 3, and so on. Since we are excluding 1, we will start our analysis from 2.

2. **Identify the Squares of Natural Numbers**:
Let's calculate the squares of the first few natural numbers:
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(4^2 = 16\)
- \(5^2 = 25\)
- \(6^2 = 36\)

3. **Check for Multiples of 3**:
We need to see if these squares are multiples of 3 or how much they exceed the nearest multiple of 3:
- For \(2^2 = 4\): The nearest multiple of 3 is 3. \(4 - 3 = 1\)
- For \(3^2 = 9\): This is a multiple of 3.
- For \(4^2 = 16\): The nearest multiple of 3 is 15. \(16 - 15 = 1\)
- For \(5^2 = 25\): The nearest multiple of 3 is 24. \(25 - 24 = 1\)
- For \(6^2 = 36\): This is a multiple of 3.

4. **Generalizing the Findings**:
From our calculations, we see that:
- The square of a natural number can either be a multiple of 3 or exceed the nearest multiple of 3 by 1.

5. **Conclusion**:
Therefore, we can conclude that the square of a natural number (other than 1) is either a multiple of 3 or exceeds a multiple of 3 by **1**.

### Final Answer:
The square of a natural number (other than 1) is either a multiple of 3 or exceeds a multiple of 3 by **1**.