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For certain positive integers a and b, the greatest common divisor of (September 2023 - G01)
For certain positive integers a and b, the greatest common divisor of a and b is 1, and 9a = 4b.
If it can be determined, which of the following statements must be true for a and b ?
F. 2 is a prime factor of a, and 3 is a prime factor of b.
G. 2 is a prime factor of a, and 3 is not a prime factor of b.
H. 2 is not a prime factor of a, and 3 is a prime factor of b.
J. 2 is not a prime factor of a, and 3 is not a prime factor of b.
K. Cannot be determined from the given information.
Solution
Correct answer: F. 2 is a prime factor of a, and 3 is a prime factor of b.
From the given information, we have:
9a=4b
a=\frac{4}{9}b
The greatest common divisor of a and b is 1, so a and b are coprime.
gcd(a,b)=1
gcd(\frac{4}{9}b,b)=1
b=9c
a=4c
gcd(4c,9c)=1
c=1
Therefore, a and b are both positive integers.
a=4
b=9
The answer is F. 2 is a prime factor of a, and 3 is a prime factor of b.
Note: The solution above assumes that the greatest common divisor of a and b is 1. If this is not the case, then the answer will be different.
Exam Edition: September 2023 - G01
Exam Year: 2023
Related topics: Number Theory, Greatest Common Divisor (GCD), Algebra