GATE CS 2013 Q62 — General Aptitude

To find the probability that a randomly selected $2$-digit number is not divisible by $7$, we can use the following steps:

Step 1: Find the total number of $2$-digit integers
The $2$-digit integers range from $10$ to $99$ inclusive.
$$\text{Total } 2\text{-digit integers } (N) = 99 - 10 + 1 = 90$$

Step 2: Find the number of $2$-digit integers divisible by $7$
The multiples of $7$ that are $2$-digit numbers are:
$$14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98$$ These can be represented as $7 \times k$, where $k$ ranges from $2$ (since $7 \times 2 = 14$) to $14$ (since $7 \times 14 = 98$).
$$\text{Number of multiples } (n) = 14 - 2 + 1 = 13$$

Step 3: Find the number of $2$-digit integers NOT divisible by $7$
$$\text{Favorable outcomes} = 90 - 13 = 77$$

Step 4: Calculate the probability
$$\text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{77}{90}$$

Hence, the correct option is D.