To determine which numbers between 2 and 9 can exactly divide 234, we will find the divisors of 234 within that range. A number divides 234 exactly if the remainder when dividing is 0, i.e., 234 ÷ divisor must be an integer.

Let's check each number between 2 and 9:

1. 234 ÷ 2 = 117 (exact, so 2 is a divisor)
2. 234 ÷ 3 = 78 (exact, so 3 is a divisor)
3. 234 ÷ 4 = 58.5 (not exact, so 4 is not a divisor)
4. 234 ÷ 5 = 46.8 (not exact, so 5 is not a divisor)
5. 234 ÷ 6 = 39 (exact, so 6 is a divisor)
6. 234 ÷ 7 = 33.43 (not exact, so 7 is not a divisor)
7. 234 ÷ 8 = 29.25 (not exact, so 8 is not a divisor)
8. 234 ÷ 9 = 26 (exact, so 9 is a divisor)

Conclusion:

The numbers between 2 and 9 that can exactly divide 234 are: 2, 3, 6, and 9.

Would you like details on any of the steps or further explanation?

Here are five questions based on this topic:

1. What are all the divisors of 234, not limited to the range between 2 and 9?
2. How would you find the greatest common divisor (GCD) of two numbers like 234 and another number?
3. How can you determine if a number is divisible by a certain divisor without performing the division?
4. What is the prime factorization of 234?
5. How can divisibility rules be applied to quickly check divisibility by numbers like 3 or 9?

Tip: When checking divisibility by 3 or 9, add the digits of the number. If the sum is divisible by 3 or 9, then the number is also divisible by that divisor.