Question

Prove that square of any positive integers of the form 5m+1 will leave a remainder 1 when divisible by 5 for some integer m

Answer 1

we know that, Positive integers are 1,2,3,4,5,.....
Square of any positive integer of the form 5m+1,
then, (5m+1)^2 = 25m^2 + 10m + 1

Here if we put the value of m = 1,2,3,4.... then we get a no. i.e. multiple of 5
for eg. take m = 2
that is, 25(2)^2 +10(2)+1 = 121.
if we divide this value by 5 than 1 will remains as a remainder.
So. By this we proved that square of any positive integer of the form 5m+1 will leave a remainder 1 when divisible by 5 for positive integer m