Question

Let a be any odd positive integer we need to prove that a is of the form 6q + 1 , or 6q + 3 , or 6q + 5 , where q is some integer. since a is an integer consider b = 6 another integer applying euclid's division lemma we get

Answer 1

Let a be any positive odd integer
b = 6 and q is any integer
Therefore
r= 0 , 1,2,3,4,5 ( since 0=r, and r is smaller than b ).
Odd integer = 1, 3,5

By Euclid division lemma
A=bq+r
For r=1
A = 6q+1
For r=3
a= 6q+3
For r=5
A = 6q +5

Hence for any positive odd integer is either of the form 6q+1 , 6q+3 or 6q+5