Use app×
Join Bloom Tuition
One on One Online Tuition
JEE MAIN 2025 Foundation Course
NEET 2025 Foundation Course
CLASS 12 FOUNDATION COURSE
CLASS 10 FOUNDATION COURSE
CLASS 9 FOUNDATION COURSE
CLASS 8 FOUNDATION COURSE
0 votes
9.6k views
in Number System by (50.2k points)
closed by

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

1 Answer

+1 vote
by (55.4k points)
selected by
 
Best answer

Let a and b be any two positive integers, such that a > b.

Then, a = bq + r, 0 ≤ r < b …(i) [by Euclid’s division lemma]

On putting b = 2 in Eq. (i), we get

a = 2q + r, 0 ≤ r < 2 …(ii)

image r = 0 or 1

When r = 0, then from Eq. (ii), a = 2q, which is divisible by 2

When r = 1, then from Eq. (ii), a = 2q + 1, which is not divisible by 2.

Thus, every positive integer is either of the form 2q or 2q + 1.

That means every positive integer is either even or odd. So, if a is a positive even integer, then a is of the form 2q and if a, is a positive odd integer, then a is of the form 2q + 1.

Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students.

Categories

...