Question

Any integers square when divided with 7 will never give remainder 5. Prove it

Answer 1

Step-by-step explanation:

Let's put the integers in the format of a=7n-m, 0≤m<7

(7n-m)²= 7n(7n-2m)+m², since the first part is the factor of 7, m²/7 will give us the remainder:

• m=0 ⇒ rem=0
• m=1 ⇒ rem =1
• m=2 ⇒ rem=4
• m=3 ⇒ rem= 9-7= 2
• m=4 ⇒ rem= 16-14= 2
• m=5 ⇒ rem= 25-21= 4
• m=6 ⇒ rem= 36- 35= 1

So we only got 1, 2, 4 as possible remainder and 5 is not one of them