Show that every positive integer is of the form 2q and that every positive odd integer is of the from 2q + 1, where q is some integer.
According to Euclid's division lemma, if a and b are two positive integers such that a is greater than b; then these two integers can be expressed as
a = bq + r; where 0 ≤ r < b
b = 2; then a = bq + r will reduce to
a = 2q + r; where 0 ≤ r < 2,
i.e., r = 0 or r = 1
If r = 0, a = 2q + r ⇒ a = 2q
i.e., a is even
and, if r = 1, a = 2q + r ⇒ a = 2q + 1
i.e., a is add; as if the integer is not even; it will be odd.
Since, a is taken to be any positive integer so it is applicable to the every positive integer that when it can be expressed as
a = 2q
∴ a is even and when it can expressed as
a = 2q + 1; a is odd.
Hence the required result.
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- Euclid’s Division Lemma