Q4.An integer is chosen at random from the first 100 positive integers. What is the probability thatthe integer is divisible by 6 or 8?

To solve this problem, we first need to find out how many numbers between 1 and 100 are divisible by 6 and 8.

Step 1: Find out how many numbers are divisible by 6. The smallest number divisible by 6 is 6 and the largest number divisible by 6 in the range 1-100 is 96. The numbers divisible by 6 form an arithmetic sequence 6, 12, 18, ..., 96. To find the total numbers in this sequence, we use the formula for the nth term of an arithmetic sequence which is a + (n - 1)d where a is the first term, d is the common difference and n is the number of terms. Solving the equation 6 + (n - 1)6 = 96, we get n = 16. So, there are 16 numbers divisible by 6.

Step 2: Find out how many numbers are divisible by 8. The smallest number divisible by 8 is 8 and the largest number divisible by 8 in the range 1-100 is 96. The numbers divisible by 8 form an arithmetic sequence 8, 16, 24, ..., 96. Using the same formula as above, solving the equation 8 + (n - 1)8 = 96, we get n = 12. So, there are 12 numbers divisible by 8.

Step 3: Find out how many numbers are divisible by both 6 and 8 (i.e., divisible by 24). The smallest number divisible by 24 is 24 and the largest number divisible by 24 in the range 1-100 is 96. The numbers divisible by 24 form an arithmetic sequence 24, 48, 72, 96. Solving the equation 24 + (n - 1)24 = 96, we get n = 4. So, there are 4 numbers divisible by 24.

Step 4: Use the principle of inclusion and exclusion. The principle of inclusion and exclusion states that for any two sets A and B, the size of their union is |A| + |B| - |A ∩ B|. Here, |A| is the number of numbers divisible by 6, |B| is the number of numbers divisible by 8, and |A ∩ B| is the number of numbers divisible by 24. So, the total number of numbers divisible by 6 or 8 is 16 + 12 - 4 = 24.

Step 5: Calculate the probability. The probability of an event is the ratio of the favorable outcomes to the total outcomes. Here, the total outcomes are 100 (since we are choosing from the first 100 positive integers) and the favorable outcomes are 24 (the numbers divisible by 6 or 8). So, the probability is 24/100 = 0.24.