Hint: According to Euclid’s Division Lemma if we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition a = bq + r where 0 ≤ r ≤ b.
As we know by Euclid's Division Lemma,
a = bq + r
where $0 \leqslant r < b$
Let positive integer be a
And b = 2
Hence a = 2q + r
where $0 \leqslant r < 2$
So, either r = 0 or r = 1
So, a = 2q or a = 2q + 1
If a is of the form 2q, then a is an even integer. Also, a positive be either even or odd.
Therefore, any positive odd integer is of form 2q + 1.
NOTE: Euclidean division can also be extended to negative dividend (or negative divisor) using the same formula. You can work out a few examples on the same.