The problem states:

1. The number is between 17 and 25.
2. The number is a multiple of 3.
3. 6 is not a factor of this number.

Step-by-step solution:

1. Find multiples of 3 between 17 and 25:

   • The multiples of 3 in this range are:
     • 18, 21, 24.

2. Check for the condition: 6 is not a factor:

   • A number is a factor of 6 if it is divisible by both 2 and 3.
   • Among the numbers above:
     • 18: Divisible by 6 (because $18 \div 6 = 3$).
     • 21: Not divisible by 6.
     • 24: Divisible by 6 (because $24 \div 6 = 4$).

3. Conclusion:

   • The only number that satisfies all conditions is 21.

Final Answer:

The number is 21.

Would you like further explanation or additional examples? Here are some follow-up questions:

1. What are other examples of numbers that are multiples of 3 but not factors of 6?
2. Can a number divisible by 2 always be a factor of 6?
3. How would this problem change if the range were between 10 and 30?
4. What are the properties of numbers that are multiples of both 2 and 3?
5. Can we generalize this to any other similar problem?

Tip: When working with factors, always verify divisibility by all components (e.g., 6 means both 2 and 3).