Derek wrote 8,9,12 and 15 on the black board. Which pairs of theses numbers are relatively prime?
my answer is 8,and 9. and 8,15.
am i right if not please tell me the answers.
I agree.
I agree.
To determine whether pairs of numbers are relatively prime, we need to find their greatest common divisor (GCD). If the GCD of a pair is 1, then the numbers are relatively prime.
Let's find the GCD of the given pairs:
1) GCD(8, 9):
To find the GCD of 8 and 9, we can use the Euclidean algorithm. Here's how it works:
- Divide 9 by 8: 9 ÷ 8 = 1 remainder 1
- Divide 8 by 1: 8 ÷ 1 = 8 remainder 0
Since the remainder is 0, we stop the process. The last nonzero remainder is 1.
Therefore, the GCD of 8 and 9 is 1.
2) GCD(8, 12):
Using the same Euclidean algorithm:
- Divide 12 by 8: 12 ÷ 8 = 1 remainder 4
- Divide 8 by 4: 8 ÷ 4 = 2 remainder 0
Again, the remainder is 0, and the last nonzero remainder is 4.
Thus, the GCD of 8 and 12 is 4.
3) GCD(8, 15):
Applying the Euclidean algorithm, we find:
- Divide 15 by 8: 15 ÷ 8 = 1 remainder 7
- Divide 8 by 7: 8 ÷ 7 = 1 remainder 1
- Divide 7 by 1: 7 ÷ 1 = 7 remainder 0
As before, the remainder is 0, and the last nonzero remainder is 1.
Hence, the GCD of 8 and 15 is 1.
Based on these calculations, you are correct in stating that the pairs (8, 9) and (8, 15) are relatively prime since their GCDs are both 1. However, the pair (8, 12) is not relatively prime since its GCD is 4.
Therefore, your answer is partially correct.