Classify each of the following statements as true or false where a and b are whole numbers. ?
a. If GCD(a,b)= 1, then a and b cannot both be even.
b. If GCD(a,b) = 2, then both a and b are even.
c. If a and b are even, then GCD(a,b) = 2.
a. true, since 2 would be a divisor of each
b. true, since both are multiples of 2
c. false. Consider GCD(12,16)=4
a. True. If the greatest common divisor (GCD) of two numbers is 1, it means that they have no common factors other than 1. Since both numbers are integers, if they were both even, they would have a common factor of 2, making the GCD greater than 1.
b. True. If the GCD of two numbers is 2, it means that both of them have 2 as a common factor. In other words, both numbers are divisible by 2, which means they are even numbers.
c. True. If both a and b are even, it means they are divisible by 2. Therefore, 2 is a common factor between them, making it the greatest common divisor.
a. True. If the gcd(a,b) = 1, it means that the only positive integer that divides both a and b is 1. Since even numbers are divisible by 2, if both a and b were even, then 2 would be a common divisor, contradicting the assumption that gcd(a,b) = 1.
b. False. If gcd(a,b) = 2, it means that both a and b are divisible by 2. However, it does not necessarily mean that both a and b are even. For example, if a = 4 and b = 6, gcd(4,6) = 2, where a is even and b is odd.
c. True. If both a and b are even, it implies that they are divisible by 2. Therefore, 2 is a common divisor of a and b. Since 2 is the largest common divisor, it follows that gcd(a,b) = 2.
To classify each statement as true or false, we need to understand the properties of greatest common divisor (GCD) and even numbers.
a. If GCD(a, b) = 1, then a and b cannot both be even.
This statement is true. By definition, if the GCD of two numbers is 1, it means they do not share any common factors other than 1. Since even numbers are divisible by 2, if both a and b were even, their GCD would be at least 2, not 1.
b. If GCD(a, b) = 2, then both a and b are even.
This statement is false. While it is true that if both a and b are even, their GCD will be at least 2, it does not guarantee that the GCD will be specifically 2. For example, if a = 4 and b = 6, both numbers are even, but their GCD is 2, not 4.
c. If a and b are even, then GCD(a, b) = 2.
This statement is true. When both a and b are even, we can express them as a = 2k and b = 2m, where k and m are integers. The GCD of a and b can be found by dividing both a and b by their largest common power of 2, which is 2 raised to the power of 1. Therefore, the GCD of a and b will be 2.
In summary:
a. True
b. False
c. True