Question

If $x \equiv y(\bmod 2)$, then which of the following is/are correct? (a) $x$ is even and $y$ is odd (b) Both $\mathrm{x}$ and $\mathrm{y}$ are odd. (c) Both $\mathrm{x}$ and $\mathrm{y}$ are even. (1) Only (c) (2) Only (a) (3) Both (b) and (c) (4) Both (a) and (b)

Name: Problem 17, Chapter 17: The IIT Foundation series of Mathematics Class
Uploaded: 2021-08-15T13:41:12-08:00
Duration: 59 s
Channel: Kumar Vaibhav
Description: If x ≡y(2), then which of the following is/are correct? (a) x is even and y is odd (b) Both x and y are odd. (c) Both x and y are even. (1) Only (c) (2) Only (a) (3) Both (b) and (c) (4) Both (a) and (b)

   If $x \equiv y(\bmod 2)$, then which of the following is/are correct?
(a) $x$ is even and $y$ is odd
(b) Both $\mathrm{x}$ and $\mathrm{y}$ are odd.
(c) Both $\mathrm{x}$ and $\mathrm{y}$ are even.
(1) Only (c)
(2) Only (a)
(3) Both
(b) and (c)
(4) Both (a) and (b)

The IIT Foundation series of Mathematics Class

Trishna Knowledge… 5th Edition

Chapter 17, Problem 17

Instant Answer

Solved by Expert Kumar Vaibhav

08/15/2021

Step 1

This means that $x$ and $y$ leave the same remainder when divided by 2. Show more…

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If $x \equiv y(\bmod 2)$, then which of the following is/are correct? (a) $x$ is even and $y$ is odd (b) Both $\mathrm{x}$ and $\mathrm{y}$ are odd. (c) Both $\mathrm{x}$ and $\mathrm{y}$ are even. (1) Only (c) (2) Only (a) (3) Both (b) and (c) (4) Both (a) and (b)

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Key Concepts

Equivalence Relations

An equivalence relation partitions a set into disjoint equivalence classes. In the context of modular arithmetic, congruence modulo n satisfies reflexivity, symmetry, and transitivity, ensuring that integers can be classified into sets where each member is equivalent to the others under the given modulus.

Modular Arithmetic

Modular arithmetic involves calculations with the remainders of integers when divided by a fixed positive number (the modulus). The notation x ? y (mod n) means that x and y have the same remainder when divided by n, which structures the integers into equivalence classes based on this remainder.

Parity

Parity distinguishes integers as either even or odd. When working modulo 2, the possible remainders are 0 and 1; thus, if two numbers are congruent modulo 2, they share the same parity. This means they are either both even or both odd.

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