Courses
Courses for Kids
Free study material
Free LIVE classes
More

Show that every positive even integer is of the form 2q and every positive odd integer is of the form 2q + 1, where q is some integer.

Last updated date: 17th Mar 2023
Total views: 208.3k
Views today: 2.87k
Answer
VerifiedVerified
208.3k+ views

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.


Complete answer: 


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.