Remainder When the Square of a Prime Number Greater Than 3 Is Divided by 6

Video Explanation

Watch the video below for a clear explanation:

Solution

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

(a) 1    (b) 3    (c) 2    (d) 4

Step 1: Observe the Nature of Prime Numbers Greater Than 3

Any prime number greater than 3 is odd and not divisible by 3.

So, such a prime number can be written in the form:

6k ± 1, where k is a natural number.

Step 2: Square the Number

(6k + 1)² = 36k² + 12k + 1

(6k − 1)² = 36k² − 12k + 1

Step 3: Find the Remainder on Division by 6

Both expressions are of the form:

6 × (some integer) + 1

So, the remainder in each case is 1.

Final Answer

✔ The remainder is 1.

✔ Correct option: (a) 1

Conclusion

Thus, the square of any prime number greater than 3 always leaves a remainder 1 when divided by 6.