Question

show that any positive integer is of the form 4q+1 and 4q+3, where q is positive integer.​

Answer 1

Answer:

let n be an positive integer.

on dividing n by 4 let q be the quotient and r be the remainder.

So by euclid's divison lemma,we have

n=4q+r,where 0greater than or equal to r and r greater than 4

therefore n=4q or (4q+1) or (4q+2) or (4q+3)

clearly 4q and (4q+2) are even since n is odd,so n not equal to 4m and n not equal to (4m+3),for some integer q.

Hence,any positive odd integer is of the form (4q+1) or (4q+3) fir some integer q.