Relation on integers is defined by :

$aRb \Leftrightarrow a - b$ is divisible by 5.
This relation is ________.
(A) Reflexive only
(B) Symmetric only
(C) Equivalence relation
(D) Neither symmetric nor transitive

Question

Relation on integers is defined by :

$aRb \Leftrightarrow a - b$ is divisible by 5.
This relation is ________.
(A) Reflexive only
(B) Symmetric only
(C) Equivalence relation
(D) Neither symmetric nor transitive

Accepted Answer

> [!TLDR]
> Divisibility of $a-b$ by 5 is congruence mod 5 — reflexive, symmetric and transitive — hence an equivalence relation.

### Step 1 — Reflexive?
For any integer $a$, $a-a = 0 = 5	imes0$ is divisible by 5, so $aRa$ holds. **Reflexive ✓**

### Step 2 — Symmetric?
If $a-b = 5k$ for some integer $k$, then $b-a = -5k = 5(-k)$ is also divisible by 5, so $bRa$. **Symmetric ✓**

### Step 3 — Transitive?
If $a-b=5k$ and $b-c=5m$, then $a-c = (a-b)+(b-c) = 5(k+m)$ is divisible by 5, so $aRc$. **Transitive ✓**

Since all three properties hold, $R$ is an equivalence relation.

**Why the other options are wrong:** (A), (B) and (D) each claim only some properties fail, but the checks above show all three hold simultaneously. …

Step 1 — Reflexive?

Step 2 — Symmetric?

Step 3 — Transitive?

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