To prove:the given statement is true or not.
Explanation of Solution
Given information:
The given statement:
“Ifa andb are relatively prime and b and c are relatively prime, then a and c are relatively prime.”
Concept used:
Two integers are relatively prime if their greatest common factor is 1
Proof:
By definition, the greatest common factor (GCF) of two integers is the greatest integers that is a factor of each.
Ifa and b are relatively prime, then their greatest common factor is 1, that is, they have no common factors except for 1
Similarly, if b and c are relatively prime, then they also have no common factors except for 1
However, it is not necessary that a and cwill also be relatively prime.
For example, 4 and 5 have no common factors, so their GCF is 1, that is they are relatively prime. Also 5 and 6 have no common factors, so their GCF is 1, that is they are relatively prime.
However, 4 and 6 have a common factor 2, so their GCF is 2. Thus, 4 and 6 are not relatively prime.
The given statement is not true.
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Algebra and Trigonometry: Structure and Method, Book 2
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