Q15 of 46 Page 1

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

According to Euclid’s Division Lemma


If a and b are two positive integers, then a = bq + r


where 0 ≤ r < b


Let a be any positive odd integer and b = 6. We apply the division algorithm with a and b = 6.


Since 0 ≤ r < 6, the possible remainders are 0, 1, 2, 3, 4, 5.


i.e. a can be 6q, 6q + 1, 6q +2, 6q + 3, 6q + 4, 6q + 5 where q is the quotient.


As we know a is odd, a can’t be 6q or 6q + 2 or 6q + 4 because they are divisible by 2.


Therefore, any positive odd integer is of the form 6q + 1 or 6q + 3 or 6q + 5.


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