Out of all the 2-digit integers between 1 and 100, a 2-digit number has to be selected at random. What is the probability that the selected number is not divisible by 7?

Steps to Solve

1. Identify the total number of 2-digit integers

The range of 2-digit integers is from 10 to 99. To find the total number of these integers, we can calculate:

$$ 99 - 10 + 1 = 90 $$

So, there are 90 2-digit integers.

2. Determine the 2-digit integers divisible by 7

We need to find the smallest and largest 2-digit multiples of 7.

• The smallest 2-digit integer divisible by 7 is $14$ (since $7 \times 2 = 14$).
• The largest 2-digit integer divisible by 7 is $98$ (since $7 \times 14 = 98$).

Next, we find how many multiples of 7 are there between 14 and 98. This can be represented by the equation:

$$ n = \frac{98 - 14}{7} + 1 $$

Calculating this gives us:

$$ n = \frac{84}{7} + 1 = 12 + 1 = 13 $$

Thus, there are 13 two-digit integers divisible by 7.

3. Calculate the number of 2-digit integers not divisible by 7

To find the number of integers that are not divisible by 7, subtract the number of integers divisible by 7 from the total number of 2-digit integers:

$$ 90 - 13 = 77 $$

4. Calculate the probability

The probability of selecting a 2-digit integer that is not divisible by 7 is the ratio of the number of integers not divisible by 7 to the total number of 2-digit integers:

$$ P = \frac{77}{90} $$

5. Simplify the probability (if necessary)

We can simplify the fraction $P = \frac{77}{90}$:

The GCD of 77 and 90 is 1, so it's already in its simplest form.