Real Numbers - Euclid's Lemma

Ex 1.1 Q2. CBSE 10th Maths NCERT exercise solution.
NCERT Solutions for Class 10 Maths | Chapter: Real Numbers | Concept: Euclid's Division Algorithm

Question 2: Show that any positive odd integer is of the form 6q + 1, or 6q + 3, or 6q + 5, where q is some integer.

Explanatory Answer

Let ‘a’ be any positive odd integer.

Let us apply Euclid’s division algorithm on ‘a’ with 6 as the divisor.
a = 6q + r, 0 ≤ r < 6
r = 0, 1, 2, 3, 4, 5
So, a = 6q + 0 or 6q + 1 or 6q + 2 or 6q + 3 or 6q + 4 or 6q + 5.
6q + 0, 6q + 2, and 6q + 4 are divisible by 2. Hence, even.
The remaining 3 numbers 6q + 1, 6q + 3 and 6q + 5 are odd.
So, any odd positive integer will be of the form 6q + 1, 6q + 3 or 6q + 5.