Que:Show that any positive odd integer is of the form 6q +1 or 6q + 3 or 6q + 5 where q is some integer By substituting b= 6 we get: a = 6q + r where[ r= 0,1,2,3,4,5] If r = 1, a= 6q +1, 6q + 1 is not divisible by 2 If r = 3, a= 6q +3, 6q + 3 is not divisible by 2 If r = 5, a= 6q +5, 6q +5 is not divisible by 2 Hence the remaining integers i.e 6q + 1, 6q + 3 and 6q + 5 are odd

Que:Use Euclid’s division lemma to show that square of any posiive integer is either of the form 3m or 3m+1 for any integer m. By substituting b= 3 we get, a = 3q + r where [ r= 0,1,2,3 ]

From (1) we know that 9q 2 is a square of the form 3m, where m=3q 2 From (3) we know that 9q 2 +12q+4 i.e 3(3q2+2q+1)+1 is a square of the form 3m+1, where m=3q 2 +4q+1 From (2) we know that 9q 2 +6q+1 i.e. 3(3q 2 +2q)+1 is a square of the form 3m+1, where m=3q 2 +2q

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