Remainder Calculation for Squared Number

This solution explains how to find the remainder when the square of a number is divided by 7, given the remainder of the original number.

Problem Statement Analysis

We are given a number that leaves a remainder of 4 when divided by 7. We need to find the remainder when the square of this number is divided by 7.

• Let the number be represented by N.
• Given: N divided by 7 leaves a remainder of 4.
• This can be written using modular arithmetic as: $N \equiv 4 \pmod{7}$.
• We need to find the remainder of $N^2$ when divided by 7, or $N^2 \pmod{7}$.

Step-by-Step Solution

1. Start with the given congruence:

   $N \equiv 4 \pmod{7}$

2. Square both sides of the congruence:

   $N^2 \equiv 4^2 \pmod{7}$

3. Calculate the square of 4:

   $N^2 \equiv 16 \pmod{7}$

4. Find the remainder when 16 is divided by 7:

   16 can be written as $2 \times 7 + 2$.

   Therefore, $16 \equiv 2 \pmod{7}$.

5. Substitute this back into the congruence for $N^2$:

   $N^2 \equiv 2 \pmod{7}$

Conclusion

The remainder when the square of the number is divided by 7 is 2.