A number when divided by 6 leaves remainder 3. When the square of the same number is divided by 6, the remainder is:

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the given information
We are told that a number, when divided by 6, leaves a remainder of 3. This can be expressed mathematically as:
\[ n \equiv 3 \ (\text{mod} \ 6) \]
This means that the number \( n \) can be written in the form:
\[ n = 6k + 3 \]
for some integer \( k \).

### Step 2: Find the square of the number
Next, we need to find the square of this number:
\[ n^2 = (6k + 3)^2 \]
Expanding this using the formula for the square of a binomial:
\[ n^2 = (6k)^2 + 2 \cdot (6k) \cdot 3 + 3^2 \]
\[ n^2 = 36k^2 + 36k + 9 \]

### Step 3: Simplify the expression
Now, we can factor out 6 from the terms in \( n^2 \):
\[ n^2 = 6(6k^2 + 6k) + 9 \]
This shows that when \( n^2 \) is divided by 6, the remainder is determined by the term \( 9 \).

### Step 4: Find the remainder when dividing by 6
Now, we need to find the remainder of \( 9 \) when divided by \( 6 \):
\[ 9 \div 6 = 1 \quad \text{(quotient)} \]
\[ 9 - (6 \cdot 1) = 3 \quad \text{(remainder)} \]

### Conclusion
Thus, when the square of the number \( n \) is divided by 6, the remainder is:
\[ \text{Remainder} = 3 \]

### Final Answer
The remainder when the square of the number is divided by 6 is **3**.

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