To solve the congruence relation $1 \equiv b \mod{7}$, we need to find values of $b$ such that when $b$ is divided by 7, it leaves a remainder of 1.

Let's check each option:

1. For $b = 6$: $6 \mod 7 = 6 \quad (\text{Not } 1)$

2. For $b = 14$: $14 \mod 7 = 0 \quad (\text{Not } 1)$

3. For $b = 13$: $13 \mod 7 = 6 \quad (\text{Not } 1)$

4. For $b = 15$: $15 \mod 7 = 1 \quad (\text{This is } 1)$

Thus, the correct value of $b$ that makes the congruence true is 15.

If you have any questions or need further details, let me know! Here are some related questions you might consider:

1. What is the general form of solutions to $1 \equiv b \mod{7}$?
2. How do you determine if a number is congruent to another modulo $n$?
3. Can you provide examples of other congruences involving different moduli?
4. What happens to congruence relations if the modulus is changed?
5. How do you solve more complex modular equations?

Tip: To quickly check congruences, always compute the modulo operation to see if the remainders match.