2. Show that four divides any perfect square or leaves a remainder 1. Also show that nine divides cube of any integer or leaves 1 or8 as remainder.sir plzzzzzzzzzzzsolvei will surely approve............

Dear student,

any number is either 2n or 2n+1.

2n^2 = 4n^2 so this is divisible by 4

(2n+1)^2 = 4n^2 + 4n + 1 

in the above term 4n^2 and 4n are divisible by 4 and leaves a reminder of 1

similarly for a cube of an interger(which can be expressed in one of these forms 3n or 3n+1 or 3n+2) divided by 9

[3n]^3 = 27n^3 when divided by 9 leaves 0

[3n+1]^3 = 27n^3 + 27n + 9n^2 + 1 when divided by 9 leaves 1 

[3n+2]^2 = 27n^3 + 36n + 18n^2 + 8 when divided by 9 leaves 8

i hope this explains !!

Approved