Question

show that any positive odd integer is of the from 6q+1, 6q+3, 6q+5 where q is some integer

Answer 1

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Answer 2

Hey friend!

To show that any positive odd integer is of the from 6q+1, 6q+3, 6q+5 where q is some integer.

Here is your answer:

By Euclid division algorithm,

We know that a = bq + r , 0 ≤ r ≤b

Let a be any positive integer and b = 6

Then, by Euclid’s algorithm, a = 6q + r for some integer q ≥ 0,

and r = 0, 1, 2, 3, 4, 5 , or 0 ≤ r <6

Therefore, a = 6q or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5

⇔ 6q + 0 : 6 is divisible by 2,

⇒ it is an even number.

⇔ 6q + 1 : 6 is divisible by 2, but 1 is not divisible by 2

⇒ it is an odd number.

⇔ 6q + 2 : 6 is divisible by 2, and 2 is divisible by 2

⇒ it is an even number.

⇔ 6q + 3 : 6 is divisible by 2, but 3 is not divisible by 2

⇒ it is an odd number.

⇔ 6q + 4 : 6 is divisible by 2, and 4 is divisible by 2

⇒ it is an even number.

⇔ 6q + 5 : 6 is divisible by 2, but 5 is not divisible by 2

⇒ it is an odd number.

Therefore,

any odd integer can be expressed in the form 6q + 1 or 6q + 3 or 6q + 5

Hence proved.

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