To determine the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] that satisfy the given conditions, we follow these steps:
1. Given Conditions:
- [tex]\(\operatorname{gcd}(a, 4) = 2\)[/tex]
- [tex]\(\operatorname{gcd}(a, b) = 1\)[/tex]
- [tex]\(4 > a > b\)[/tex]
- [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are natural numbers
2. Analyzing the Conditions:
- Since [tex]\(\operatorname{gcd}(a, 4) = 2\)[/tex], [tex]\(a\)[/tex] must be a divisor of 4 and share a greatest common divisor of 2 with 4, meaning [tex]\(a\)[/tex] can only be 2 (as 2 is the only divisor of 4 between 1 and 4 that makes [tex]\(\operatorname{gcd}(a, 4) = 2\)[/tex]).
- [tex]\(\operatorname{gcd}(a, b) = 1\)[/tex] implies [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are coprime (they share no common factors other than 1).
3. Finding Suitable Values:
- Let's consider [tex]\(a = 2\)[/tex] since it satisfies [tex]\(\operatorname{gcd}(a, 4) = 2\)[/tex].
- We now need to find [tex]\(b\)[/tex] such that [tex]\(4 > 2 > b\)[/tex] and [tex]\(\operatorname{gcd}(2, b) = 1\)[/tex].
4. Possible Values for [tex]\(b\)[/tex]:
- Since [tex]\(b < 2\)[/tex] and [tex]\(b\)[/tex] is a natural number, the only possible value for [tex]\(b\)[/tex] is 1.
- Check the condition [tex]\(\operatorname{gcd}(2, 1) = 1\)[/tex]. This is true because 2 and 1 are coprime.
5. Verification:
- [tex]\(a = 2\)[/tex] and [tex]\(b = 1\)[/tex] satisfy:
- [tex]\(4 > a = 2 > b = 1\)[/tex]
- [tex]\(\operatorname{gcd}(2, 4) = 2\)[/tex]
- [tex]\(\operatorname{gcd}(2, 1) = 1\)[/tex]
Therefore, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are [tex]\(2\)[/tex] and [tex]\(1\)[/tex] respectively.
So the answer is [tex]\((a, b) = (2, 1)\)[/tex].
