If x and y are positive integers, each of the following could be the greatest common divisor of 30x and 15y EXCEPTa)30x.b)15y.c)15(x + y).d)15(x - y).e)15,000.Correct answer is option 'C'. Can you explain this answer?

Question

Answer

To determine the option that could NOT be the greatest common divisor (GCD) of 30x and 15y, we need to analyze each option:

A: The greatest common divisor (GCD) of 30x and 30x is 30x itself. Therefore, A could be the GCD of 30x and 15y.

B: The GCD of 15y and 15y is 15y itself. Hence, B could be the GCD of 30x and 15y.

C: The GCD of 15(x + y) can be found by factoring out the common factor of 15. We have GCD(15(x + y), 15) = 15. Therefore, C could also be the GCD of 30x and 15y.

D: The GCD of 15(x - y) can be found by factoring out the common factor of 15. We have GCD(15(x - y), 15) = 15. Hence, D could be the GCD of 30x and 15y.

E: The GCD of 15,000 and any number is 15, as 15 is a factor of 15,000. Therefore, E could be the GCD of 30x and 15y.

Based on the analysis, the option that could NOT be the GCD of 30x and 15y is option C.

Therefore, the answer is C.

Answer

Understanding the Problem
To solve which option cannot be the greatest common divisor (GCD) of $30x$ and $15y$, we first analyze the prime factorization of these expressions.
- $30x = 2 \times 3 \times 5 \times x$
- $15y = 3 \times 5 \times y$
The GCD of $30x$ and $15y$ can be expressed as:
- $GCD(30x, 15y) = 3 \times 5 \times GCD(2 \times x, y)$
This shows that the GCD is always a multiple of $15$.
Evaluating the Options
Now, let’s evaluate each option:
- a) $30x$: This can be the GCD since it is a multiple of $15$ and depends on $x$.
- b) $15y$: Similar to option a, this can also be the GCD as it is a multiple of $15$ and depends on $y$.
- c) $15(x + y)$: This expression is problematic. The GCD $GCD(30x, 15y)$ must divide both $30x$ and $15y$. However, $x + y$ is not guaranteed to be a divisor of both $x$ and $y$. Thus, $15(x + y)$ cannot be the GCD.
- d) $15(x - y)$: This can be a valid GCD if $x$ and $y$ are structured such that $x - y$ is a common factor of both $x$ and $y$.
- e) $15,000$: This is a large multiple of $15$ and can be a GCD if $x$ and $y$ are chosen appropriately.
Conclusion
The option that cannot be the GCD of $30x$ and $15y$ is c) $15(x + y)$, as it does not satisfy the properties of a GCD based on the values of $x$ and $y$.

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