Show that any positive odd integer is of form 6q 1 or 6q 3or6q 5 where q is some integer?

Question

Show that any positive odd integer is of form 6q 1 or 6q  3or6q  5 where q is some integer?

Answer

Let a be a given integer.

On dividing a by 6 , we get q as the quotient and r as the remainder such that

a = 6q + r, r = 0,1,2,3,4,5

when r=0

a = 6q,even no

when r=1

a = 6q + 1, odd no

when r=2

a = 6q + 2, even no

when r = 3

a=6q + 3,odd no

when r=4

a=6q + 4,even no

when r=5,

a= 6q + 5 , odd no

Any positive odd integer is of the form 6q+1,6q+3 or 6q+5.

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Answer

Proof that any positive odd integer is of form 6q+1 or 6q+3 or 6q+5 where q is some integer

Introduction: We need to prove that any positive odd integer is of form 6q+1 or 6q+3 or 6q+5 where q is some integer. In other words, we need to show that any odd integer can be expressed in one of these three forms.

Method of proof: We will use the method of contradiction to prove this statement. We will assume that there exists an odd integer which cannot be expressed in one of these three forms, and then we will show that this assumption leads to a contradiction.

Assumption: Let n be an odd integer which cannot be expressed in one of the three forms 6q+1, 6q+3, or 6q+5. This means that n cannot be written in the form of 6q+r where r is either 1, 3, or 5.

Proof: We can now consider the remainder of n when it is divided by 6. There are only three possibilities for this remainder: 1, 3, or 5. We will consider these three cases separately.

Case 1: When n leaves a remainder of 1 when divided by 6. This means that n can be written in the form of 6q+1. But we have assumed that there exists an odd integer which cannot be expressed in this form, so this case leads to a contradiction.

Case 2: When n leaves a remainder of 3 when divided by 6. This means that n can be written in the form of 6q+3. But we have assumed that there exists an odd integer which cannot be expressed in this form, so this case also leads to a contradiction.

Case 3: When n leaves a remainder of 5 when divided by 6. This means that n can be written in the form of 6q+5. But we have assumed that there exists an odd integer which cannot be expressed in this form, so this case also leads to a contradiction.

Conclusion: Since all three cases lead to a contradiction, our assumption that there exists an odd integer which cannot be expressed in the form of 6q+1, 6q+3, or 6q+5 must be false. Therefore, we have proved that any positive odd integer is of the form 6q+1, 6q+3, or 6q+5 where q is some integer.