Find six different numbers such that the GCF of each pair of these numbers is greater than 1 and the GCF of all the numbers is 1.

1 answer

To find six distinct numbers such that the GCF of every pair is greater than 1 and the GCF of all six numbers is 1, we can construct the numbers using prime numbers in a systematic way.

Let's consider the following six numbers, each consisting of distinct prime factors:

  1. \( a = 6 = 2 \times 3 \)
  2. \( b = 10 = 2 \times 5 \)
  3. \( c = 14 = 2 \times 7 \)
  4. \( d = 15 = 3 \times 5 \)
  5. \( e = 21 = 3 \times 7 \)
  6. \( f = 35 = 5 \times 7 \)

Check Pairwise GCF

We will check the GCF for each pair of these numbers:

  • \( \text{GCF}(a, b) = \text{GCF}(6, 10) = 2 \)

  • \( \text{GCF}(a, c) = \text{GCF}(6, 14) = 2 \)

  • \( \text{GCF}(a, d) = \text{GCF}(6, 15) = 3 \)

  • \( \text{GCF}(a, e) = \text{GCF}(6, 21) = 3 \)

  • \( \text{GCF}(a, f) = \text{GCF}(6, 35) = 1 \)

  • \( \text{GCF}(b, c) = \text{GCF}(10, 14) = 2 \)

  • \( \text{GCF}(b, d) = \text{GCF}(10, 15) = 5 \)

  • \( \text{GCF}(b, e) = \text{GCF}(10, 21) = 1 \)

  • \( \text{GCF}(b, f) = \text{GCF}(10, 35) = 5 \)

  • \( \text{GCF}(c, d) = \text{GCF}(14, 15) = 1 \)

  • \( \text{GCF}(c, e) = \text{GCF}(14, 21) = 7 \)

  • \( \text{GCF}(c, f) = \text{GCF}(14, 35) = 7 \)

  • \( \text{GCF}(d, e) = \text{GCF}(15, 21) = 3 \)

  • \( \text{GCF}(d, f) = \text{GCF}(15, 35) = 5 \)

  • \( \text{GCF}(e, f) = \text{GCF}(21, 35) = 7 \)

Check GCF of All Six Numbers

Now we need to confirm that the GCF of all six numbers is 1.

The prime factors involved are:

  • 2 from 6, 10, and 14
  • 3 from 6, 15, and 21
  • 5 from 10, 15, and 35
  • 7 from 14, 21, and 35

No prime factor can be taken from all six numbers, which means:

\[ \text{GCF}(6, 10, 14, 15, 21, 35) = 1 \]

Conclusion

Thus, the six numbers \( 6, 10, 14, 15, 21, 35 \) meet the requirements:

  • The GCF of each pair is greater than 1.
  • The GCF of all six numbers is 1.

So, one valid solution is:

Final Set of Numbers: \( 6, 10, 14, 15, 21, 35 \).