If two numbers are each divided by the same divisor, then the remainders are 6 and 7, respectively. If the sum of the two numbers be divided by the same divisor, then the remainder is 5. The divisor is:

The correct answer is Option 8.

Step-by-Step Explanation:

Let the two numbers be $x$ and $y$, and let the common divisor be $d$.

When a number is divided by a divisor, it can be written using the division algorithm as:
$\text{Dividend}=(\text{Divisor}\×\text{Quotient})+\text{Remainder}$

According to the given conditions:
1. For the first number $x$, let the quotient be $q_1$ and the remainder is 6:
$x=d{q}_1+6$

2. For the second number $y$, let the quotient be $q_2$ and the remainder is 7:
$y=d{q}_2+7$

Now, let us find the sum of the two numbers $x+y$:
$x+y=(d{q}_1+6)+(d{q}_2+7)$
$x+y=d(q_1+q_2)+13$

We are given that when the sum $x+y$ is divided by the same divisor $d$, the remainder is 5.

Notice that the term $d(q_1+q_2)$ is completely divisible by $d$. Therefore, the remainder when $x+y$ is divided by $d$ is equal to the remainder when the sum of individual remainders (13) is divided by $d$.

Hence,
$13=d\×q+5$ for some integer quotient $q$.
$d\×q=13-5=8$

Since the remainders 6 and 7 must be strictly less than the divisor $d$, we know that $d\>7$. The only factor of 8 greater than 7 is 8 itself.

Thus, the divisor is 8.