Question

if n= 5 q + 4 then check whether n×n -1 is divisible by 5 for some integer q.

Answer 1

n = 5q +4 

n*n-1 = (5q +4) (5q +4) -1

         = 25q^2 + 40q + 16 -1
         = 25q^2 +  40q + 15
     
by taking the common factor out,

        5 (5q^2 + 8q +3)

So it's a multiple of 5. 

therefore it is divisible by 5 for all integers q.

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Answer 2

Answer:

n=5q+4

n2-1=(5q+4)(5q+4)-1

n2-1=25q2+40q+16-1

n2-1=25q2+40q+15

now,

factorize 25q2+40q+15

=5(5q2+8q+3)

this is the multiple of 5

so,

it is divisible by 5

hence checked