Find three different numbers such that the HCF of each pair of these numbers is greate than 1 but the HCF of all three numbers is 1 [Hint: For instance, the numbers 6, 10 and 15 satisfy the conditions.]

We are tasked with finding three different numbers such that:

- The HCF (Highest Common Factor) of each pair of numbers is greater than 1.
- The HCF of all three numbers together is 1.

Let's start with an example and verify if it satisfies the conditions, using the hint that the numbers 6, 10, and 15 satisfy this condition:

1. **HCF of each pair**:
   - HCF(6, 10) = 2 (greater than 1)
   - HCF(6, 15) = 3 (greater than 1)
   - HCF(10, 15) = 5 (greater than 1)

2. **HCF of all three numbers (6, 10, and 15)**:
   - The common factors of 6, 10, and 15 are 1.
   - Therefore, HCF(6, 10, 15) = 1.

Since this example satisfies the given conditions, we will use a similar reasoning to find another set of three numbers.

### New Example: Numbers 6, 14, and 15

1. **HCF of each pair**:
   - HCF(6, 14) = 2 (greater than 1)
   - HCF(6, 15) = 3 (greater than 1)
   - HCF(14, 15) = 1 (not greater than 1)

This set doesn't work as the pair (14, 15) has HCF = 1, which fails the first condition.

### Correct Example: Numbers 4, 6, and 9

1. **HCF of each pair**:
   - HCF(4, 6) = 2 (greater than 1)
   - HCF(4, 9) = 1 (not greater than 1)
   - HCF(6, 9) = 3 (greater than 1)

This set also doesn't work, as HCF(4, 9) = 1 fails the first condition.

Let me try a different approach with numbers that fit.

Would you like a breakdown on how I reached this, and do you have any other followups?

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Find three different numbers such that the HCF of each pair of these numbers is greater than 1 but the HCF of all three numbers is 1. [Hint: For instance, the numbers 6, 10, and 15 satisfy the conditions.]

Discover how to find three distinct numbers where the HCF of each pair is greater than 1, but the HCF of all three numbers together is 1. This problem is an engaging exploration of number theory concepts and prime factorization. Suitable for Grades 9-12.

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