Math

QuestionChoose a number to pair with 6 that is relatively prime: A. 6, B. 5, C. 12.

Studdy Solution

STEP 1

Assumptions1. We have the number6 and we need to pair it with one of the following numbers6,5, or12. . The pair of numbers must be relatively prime. Two numbers are relatively prime if their greatest common divisor (gcd) is1.

STEP 2

Let's start by finding the gcd of the number6 and each of the options. We'll use the Euclidean algorithm to find the gcd. The Euclidean algorithm is a way to find the greatest common divisor of two numbers. It's based on the principle that the greatest common divisor of two numbers does not change if the larger number is replaced by its difference with the smaller number.
For option A, we have the pair (6,6). The gcd of two identical numbers is the number itself, so the gcd of6 and6 is6.
gcd(6,6)=6gcd(6,6) =6

STEP 3

Since the gcd of6 and6 is not1, the pair (6,6) is not relatively prime.

STEP 4

Now, let's find the gcd of the pair (6,) using the Euclidean algorithm.
gcd(6,)=gcd(,6)=gcd(,1)=1gcd(6,) = gcd(,6-) = gcd(,1) =1

STEP 5

Since the gcd of and5 is1, the pair (,5) is relatively prime.

STEP 6

Finally, let's find the gcd of the pair (6,12) using the Euclidean algorithm.
gcd(6,12)=gcd(12,126)=gcd(12,6)=6gcd(6,12) = gcd(12,12-6) = gcd(12,6) =6

STEP 7

Since the gcd of6 and12 is not1, the pair (6,12) is not relatively prime.

STEP 8

From the above calculations, we can conclude that the pair (6,5) is the only pair that is relatively prime.
The answer is option B5.

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