Question

A number when divided by 7 leaves a remainder of 4. What is the remainder when its square is divided by 7? 
 

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

In modular arithmetic, reduce each term modulo $m$ early to simplify calculations.

Answer 1

The Correct Option is B

- Step 1: Express the number in modular form - Let $n = 7k + 4$ for some integer $k$. 
- Step 2: Square the expression - \[ n^2 = (7k + 4)^2 = 49k^2 + 56k + 16 \] 
- Step 3: Take modulo 7 - Since $49k^2$ and $56k$ are multiples of 7, they leave remainder 0. Thus: \[ n^2 \equiv 16 \pmod{7} \] 
- Step 4: Simplify remainder - $16 \div 7 = 2$ remainder $2$, so: \[ n^2 \equiv 2 \ (\text{mod } 7) \] 
- Step 5: Conclusion - The remainder is $2$, matching option (2).