Question

The remainder when the square of any prime number greater than 3 is divided by 6 is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

The Correct Option is A

All prime numbers greater than 3 can be expressed in one of two forms:

\( p = 6k + 1 \)    or    \( p = 6k - 1 \)
(where \( k \) is a positive integer)

 

Case 1: \( p = 6k + 1 \)

\[ \begin{align*} p^2 &= (6k + 1)^2 \\ &= 36k^2 + 12k + 1 \\ &= 6(6k^2 + 2k) + 1 \end{align*} \]

When divided by 6, the remainder is 1.

Case 2: \( p = 6k - 1 \)

\[ \begin{align*} p^2 &= (6k - 1)^2 \\ &= 36k^2 - 12k + 1 \\ &= 6(6k^2 - 2k) + 1 \end{align*} \]

When divided by 6, the remainder is again 1.

Verification with Examples

\( 5^2 = 25 \quad \Rightarrow \quad 25 \div 6 \) leaves remainder 1

\( 7^2 = 49 \quad \Rightarrow \quad 49 \div 6 \) leaves remainder 1

\( 11^2 = 121 \quad \Rightarrow \quad 121 \div 6 \) leaves remainder 1

Conclusion

For any prime number \( p > 3 \), \( p^2 \equiv 1 \pmod{6} \).
The correct answer is (1) 1.