Question

Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

Answer 1

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let 'a' be a positive integer, b=6
Applying Euclid division Algorithm 
a=bq+r {0 _< r <  b }
a=6q+r 
now, r = 0,1,2,3,4,5
r=0,a=6q=0=6q
r=1,a=6q+1
r=2,a=6q+2
r=3,a=6q+3
r=4,a=6q+4
r=5,a=6q+5
Now,
a=6q=even 
a=6q+1=odd
a=6q+2=even
a=6q+3=odd
a=6q+4=even 
a=6q+5=odd
therefore. positive odd number is in the form of  6q+1, 6q+3, 6q+5.

Answer 2

Let a be any odd positive integer and b=6

Then, there exist integers q and r such that

a = 6q + r , 0≤r<6
a = 6q or,
a = 6q + 1 or,
a = 6q + 2 or,
a = 6q + 3 or,
a = 6q + 4 or,
a = 6q + 5


But , 6q , 6q + 2 and 6q + 4 are even positive integers

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