We need to determine for which values of $\alpha \in \mathbf N$ the relation

on $\mathbf N$ is an equivalence relation.

An equivalence relation must be:

1. Reflexive
2. Symmetric
3. Transitive

We analyze these one by one modulo $7$.

1. Rewrite the relation modulo $7$

The condition $3x+\alpha y \text{ is a multiple of }7$ means $3x+\alpha y \equiv 0 \pmod 7.$ So $(x,y)\in R \iff 3x+\alpha y\equiv 0\pmod 7.$

2. Reflexivity

For reflexivity, we need $(x,x)\in R$ for every $x\in \mathbf N$.

So: $3x+\alpha x \equiv 0 \pmod 7 \quad \forall x$ $x(3+\alpha)\equiv 0 \pmod 7 \quad \forall x.$ For this to hold for all natural numbers $x$, we must have $3+\alpha \equiv 0 \pmod 7.$ Hence, $\alpha \equiv -3 \equiv 4 \pmod 7.$

So reflexivity requires $\boxed{\alpha \equiv 4 \pmod 7}.$

3. Symmetry

Assume $(x,y)\in R$. Then $3x+\alpha y \equiv 0 \pmod 7.$ For symmetry, we must also have $(y,x)\in R$, i.e. $3y+\alpha x \equiv 0 \pmod 7.$

Now from reflexivity condition, let $\alpha \equiv 4 \pmod 7.$ Then the relation becomes $3x+4y\equiv 0 \pmod 7.$ But since $4\equiv -3 \pmod 7$, $3x+4y \equiv 3x-3y = 3(x-y) \pmod 7.$ So $(x,y)\in R \iff 3(x-y)\equiv 0 \pmod 7.$ Since $3$ is invertible modulo $7$, $x-y\equiv 0 \pmod 7,$ i.e. $x\equiv y \pmod 7.$ This is clearly symmetric.

So symmetry holds when $\alpha\equiv 4\pmod 7$.

4. Transitivity

Again, with $\alpha\equiv 4\pmod 7$, we found $(x,y)\in R \iff x\equiv y \pmod 7.$ Congruence modulo $7$ is transitive. Hence transitivity also holds.

5. Necessity and sufficiency

We have shown:

• Reflexivity forces $\alpha\equiv 4\pmod 7$.
• For this value, the relation becomes $x\equiv y\pmod 7$, which is an equivalence relation.

Therefore, $R \text{ is an equivalence relation } \iff \alpha\equiv 4\pmod 7.$

That means: $\boxed{\text{4 is the remainder when } \alpha \text{ is divided by }7}.$

6. Check options

• A: $\alpha=14$
  Then $\alpha\equiv 0\pmod 7$, not $4$. Incorrect.

• B: $\alpha$ is a multiple of $4$
  Not sufficient. Example: $\alpha=8\equiv 1\pmod 7$, fails. Incorrect.

• C: $4$ is the remainder when $\alpha$ is divided by $10$
  This means $\alpha\equiv 4\pmod{10}$, which does not necessarily imply $\alpha\equiv 4\pmod 7$. Incorrect.

• D: $4$ is the remainder when $\alpha$ is divided by $7$
  Exactly the required condition. Correct.

Final Answer

The relation is an equivalence relation if and only if $\boxed{\alpha\equiv 4\pmod 7}.$ Hence the correct option is $\boxed{\text{D}}.$