Question

show that any positive odd integer is of the form4q+1 or 4q+3 where q is some integer.
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Answer 1

Answer:

Let a be any odd positive integer and b=4

By Euclid division lemma

a=bq+r, 0<r<4 where r=0,1,2,3

a=4q+r

a=4q,

a=4q+1,

a=4q+2,

a=4q+3,

The given statement is a should be odd positive integer

So a≠4q,a≠4q+2 which is an even integer

a=4q+1 and a=4q+3

Therefore Any odd integer is of the form 4q+1,4q+3