A number is selected at random from the integers 1 to 25 (inclusive). What is the probability that the chosen number is divisible by 4 or by 7?

Introduction / Context:
Probability questions that involve divisibility over a finite set of integers are standard problems in quantitative aptitude exams. In this question, a number is picked at random from 1 to 25, and we are asked to find the probability that it is divisible by 4 or by 7. This tests understanding of basic counting, divisibility rules, and the idea of favourable outcomes over total outcomes.

Given Data / Assumptions:

• The integers under consideration are 1, 2, 3, ..., 25 (inclusive).
• Each integer from 1 to 25 is equally likely to be selected.
• We want numbers divisible by 4 or by 7.
• Standard probability formula is used: probability = favourable outcomes / total outcomes.

Concept / Approach:
We first count the total number of integers from 1 to 25. Then we count how many of those integers are divisible by 4, how many are divisible by 7, and check whether any number is divisible by both 4 and 7. The probability is then the number of distinct favourable integers divided by 25. Since 4 and 7 are relatively prime, we only need to check whether any common multiple like 28 falls inside the range, which it does not, so there is no overlap to subtract.

Step-by-Step Solution:
Total integers from 1 to 25 inclusive = 25. Multiples of 4 in this range: 4, 8, 12, 16, 20, 24. Number of multiples of 4 = 6. Multiples of 7 in this range: 7, 14, 21. Number of multiples of 7 = 3. Common multiples of 4 and 7 would be multiples of 28. Since 28 is greater than 25, there is no common multiple in the range. Therefore, total favourable integers divisible by 4 or 7 = 6 + 3 = 9. Probability = favourable outcomes / total outcomes = 9 / 25.

Verification / Alternative check:
As a quick check, we can list all favourable numbers explicitly: 4, 7, 8, 12, 14, 16, 20, 21, 24. Counting them gives 9 numbers. Since the total possibilities are 25 equally likely outcomes, the fraction 9/25 is consistent and cannot be simplified further, so it is in lowest terms.

Why Other Options Are Wrong:
3/25: This would mean only 3 favourable numbers, which is too small and does not match the actual list of multiples.
1/25: This would correspond to exactly one favourable number, which is clearly not true here.
7/25: This assumes only 7 favourable numbers, which does not match the actual count of 9.
None of these: This is incorrect because 9/25 is a valid and correct probability among the given options.

Common Pitfalls:
A common mistake is to forget to check for overlaps when using the phrase “divisible by 4 or 7” and directly add the counts even when there is a common multiple. Another frequent error is miscounting multiples in the given range or accidentally including a number outside the range. Candidates may also confuse “or” with “and” and try to find numbers divisible by both 4 and 7, which is not what the question asks.

Final Answer:
Therefore, the required probability that the selected number is divisible by 4 or by 7 is 9/25.