We know that when an integer n is divided by 5. there is a remainder in {0.1,2,3,4}. That is. there Is an integer 1 so that n = 5q + r. where r {0.1,2,3,4). By considering four cues, show that if n is not a multiple of 5, then n2 is not a multiple of 5. Note that if n is not a multiple of 5, then the remainder r {1.2.3.4}. Solution case - 1: n = 5q+1 , where q is an integer squaring on both sides => n2 = (5q+1)2 = 25q2 + 10q + 1 => n2 = 5(5q2 + 2q) + 1 Hence n2 is not multiple of 5 as it leaves remainder 1 if divided by 5 case - 2: n = 5q+2 , where q is an integer squaring on both sides => n2 = (5q+2)2 = 25q2 + 20q + 4 => n2 = 5(5q2 + 4q) + 4 Hence n2 is not multiple of 5 as it leaves remainder 4 if divided by 5 case - 3: n = 5q+3 , where q is an integer squaring on both sides => n2 = (5q+3)2 = 25q2 + 30q + 9 => n2 = 5(5q2 + 6q + 1) + 4 Hence n2 is not multiple of 5 as it leaves remainder 4 if divided by 5 case - 4: n = 5q+4 , where q is an integer squaring on both sides => n2 = (5q+4)2 = 25q2 + 40q + 16 => n2 = 5(5q2 + 8q + 3) + 1 Hence n2 is not multiple of 5 as it leaves remainder 1 if divided by 5 So,we can say that if n is not a multiple of 5,then n2 is not a multiple of 5.