Question numbers 13 to 22 carry 3 marks each.13. Use Euclid division lemma to show that the square of any positive integer cannot be of the form 5mm + 2or 5m +3 for some integer m.

Question numbers 13 to 22 carry 3 marks each.13. Use Euclid division l

1.	Question numbers 13 to 22 carry 3 marks each.13. Use Euclid division lemma to show that the square of any positive integer cannot be of the form 5mm + 2or 5m +3 for some integer m.
Answer» Let abe any positive integer.By Euclid's division lemma,a=bm+rwhere b= 5⇒a= 5m+rSo, rcan be any of 0, 1, 2, 3, 4∴ a= 5mwhen r= 0a= 5m+ 1 when r= 1a= 5m+ 2 when r= 2a= 5m+ 3 when r= 3a= 5m+ 4 when r= 4So, "a" is any positive integer in the form of 5m, 5m+ 1 , 5m+ 2 , 5m+ 3 , 5m+ 4 for some integerm.Case I :a= 5m⇒ a2= (5m)2= 25m2⇒ a2= 5(5m2)= 5q,where q= 5m2Case II :a= 5m+ 1 ⇒ a2= (5m+ 1)2= 25m2+ 10m+ 1 ⇒ a2= 5 (5m2+ 2m) + 1= 5q+ 1, whereq= 5m2+ 2mCase III :a= 5m+ 2⇒ a2= (5m+ 2)2= 25m2+ 20m+4= 25m2+ 20m+4= 5 (5m2+ 4m) + 4= 5q+ 4 whereq= 5m2+ 4mCase IV:a= 5m+ 3⇒a2=(5m+ 3)2= 25m2+ 30m+ 9=25m2+ 30m+ 5 + 4=5 (5m2+ 6m+ 1) + 4=5q+ 4 where q= 5m2+ 6m+ 1Case V: a= 5m+ 4⇒a2=(5m+ 4)2= 25m2+ 40m+ 16=25m2+ 40m+ 15 + 1=5 (5m2+ 8m+ 3) + 1=5q+ 1 where q= 5m2+ 8m+ 3From all these cases, it is clear that square of any positive integer can not be of the form 5m+ 2 or 5m+ 3