Question

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

• A. 1
• B. 2
• C. 3
• D. 4

Hint:

Use modular arithmetic to simplify remainder calculations for squares.

Answer 1

The Correct Option is B

We need the remainder when the square of the number is divided by 7.
- Step 1: Express the number. Number = \( 7k + 4 \), since it leaves remainder 4 when divided by 7.
- Step 2: Square the number.
\[ (7k + 4)^2 = 49k^2 + 56k + 16 \] - Step 3: Take modulo 7.
- \( 49k^2 = 7 \times 7k^2 \equiv 0 \pmod{7} \).
- \( 56k = 7 \times 8k \equiv 0 \pmod{7} \).
- \( 16 \div 7 = 2 \text{ remainder } 2 \Rightarrow 16 \equiv 2 \pmod{7} \).
\[ (7k + 4)^2 \equiv 2 \pmod{7} \] - Step 4: Verify. Test \( k = 0 \): Number = 4. Square = \( 16 \div 7 = 2 \text{ remainder } 2 \).
- Step 5: Check options.
- (a) 1: Incorrect.
- (b) 2: Correct.
- (c) 3: Incorrect.
- (d) 4: Incorrect.
- Step 6: Alternative check. Try numbers: 4, 11, 18. All give remainder 2 for square.
Thus, the answer is b.