The question asks for the number of integers between 1 and 100 that are divisible by both 4 and 9. Since 4 and 9 are relatively prime (their greatest common divisor is 1), a number is divisible by both if and only if it's divisible by their product, which is 4 * 9 = 36.
Therefore, we are looking for multiples of 36 between 1 and 100. These multiples are:
- 36 * 1 = 36
- 36 * 2 = 72
- 36 * 3 = 108 (This is greater than 100, so we stop here)
Thus, there are only two numbers between 1 and 100 divisible by both 4 and 9, namely 36 and 72.
The reference provided, "...there are 36–2=34 numbers between 1 and 100 divisible by 4 or 9," discusses the number of integers divisible by either 4 or 9, not by both. It is an example of using inclusion-exclusion principle, which is not relevant to the current question. It incorrectly identifies 36 numbers divisible by 4 or 9.
| Divisible by 4 | Divisible by 9 | Divisible by both 4 and 9 |
|---|---|---|
| 25 | 11 | 2 |
Answer: There are 2 numbers between 1 and 100 that are divisible by both 4 and 9.
