# Show that any positive odd integer is of the form 8q+1, 8q+3, 8q+5 or 8q+7, where q is some integer.

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by jayChad

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by jayChad

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According to division algorithm,

we can write any number ‘a’ in the form

a = 8q + r

where q is any integer and 0 <= r <= 7. So r can be 0, 1, 2, 3, 4, 5, 6 or 7.

Thus, a can be written as

a = 8q

a = 8q+2

a = 8q+3

a = 8q+4

a = 8q+5

a = 8q+6

a = 8q+7

We need only odd numbers. Since 8q, 8q+2, 8q+4, and 8q+6 are divisible by 2, they are even numbers.

So any odd integer can be written as any one of the remaining forms which are (8q+1, 8q+3, 8q+5 or 8q+7.)