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

see full answer

Start JEE / NEET / Foundation preparation at rupees 99/day !!

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

answer is A.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

We have to find the remainder when the square of any prime number greater than 3 is divided by 6.
Use the following equations:

{(a - b)}^{2} = a^{2} + b^{2} - 2 ab

{(a + b)}^{2} = a^{2} + b^{2} + 2 ab

Thus,

{(6 k - 1)}^{2} = {(6 k)}^{2} - 2 (6 k) (1) + 1^{2}

\Rightarrow {(6 k - 1)}^{2} = {36 k}^{2} - (12 k) + 1

\Rightarrow {(6 k - 1)}^{2} = {6 \times (6 k}^{2} - 2 k) + 1

Similarly:

{(6 k + 1)}^{2} = {(6 k)}^{2} + 2 (6 k) (1) + 1^{2}

\Rightarrow {(6 k + 1)}^{2} = {36 k}^{2} + (12 k) + 1

\Rightarrow {(6 k + 1)}^{2} = {6 \times (6 k}^{2} + 2 k) + 1

Both

{(6 k}^{2} - 2 k)

and

{(6 k}^{2} + 2 k)

will be natural numbers as k is a natural number.
So, when divided by 6, the constant term will be the remainder and it is equal to 1 in both cases.
Therefore, the remainder when the square of any prime number greater than 3 is divided by 6 is 1.
Hence, option 1 is correct.

Watch 3-min video & get full concept clarity

tricks from toppers of Infinity Learn

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Books for IIT JEE

Best Books for NEET

🔥 Start Your JEE/NEET Prep at Just ₹1999 / month - Limited Offer! Check Now!

Get Expert Academic Guidance – Connect with a Counselor Today!

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Books for IIT JEE

Best Books for NEET