Question

show that any +ve odd integer is in the form of 6p + 1 , 6p + 3 , 6p + 5 where p is some integer

Answer 1

Let a be any positive integer
b = 6
r = 0,1,2,3,4,5
q and r are any positive integers
∴ By Euclid's Division Lemma,
a = bq + r
In 1st case,
a = 6q (even)

In 2nd case,
a = 6q + 1 = 6p + 1 (Odd)

In 3rd case,
a = 6q + 2 (Even)

In 4th Case,
a = 6q + 3 = 6p + 3 (Odd)

In 5th Case,
a= 6q + 4 (Even)

In 6th Case,
a = 6q + 5 = 6p + 5 (Odd)

∴ All odd +ve integers are of the form 6p + 1, 6p +3 and 6p + 5.