Question

Show that any positive even integer is of the form 8p , 8p+2 , 8p+4 and 8p+6 where p is some integer.

Answer 1

Let a be any positive integer and b = 8. Then, by Euclid’s lemma a = 8q + r,

0 ≤ r < 8 and so the possible remainders are 0, 1, 2, 3, 4 , 5 , 6 and 7

That is, a can be 8q or 8q + 1 or 8q + 2 or 8q + 3 or 8q + 4 or 8q + 5 or 8q+ 6 or 8q+ 7 , where q is the quotient.

If a = 8q + 1 or 8q + 3 or 8q + 5 or 8q +7 then a is an odd integer.

And,
if a = 8q or 8q + 2 or 8q + 4 or 8q + 6 then a is an even integer.

Also, an integer can either be even or odd.

∴ Any even integer is of the form 8q or 8q + 2 or 8q + 4 or 8q + 6 , where q is some integer. ( Hence proved )