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If c and d are positive integers and m is the greatest common factor of c and d, then m must be the greatest common factor of c and which of the following integers?
- a)c + d
- b)2 + d
- c)cd
- d)2d
- e)d^2
Correct answer is option 'A'. Can you explain this answer?
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If c and d are positive integers and m is the greatest common factor o...
A. c + d:
The GCF of c and d may or may not divide c + d evenly. For example, if c = 2 and d = 4, their GCF is 2, but 2 does not divide 2 + 4 evenly. Therefore, we cannot conclude that the GCF of c and d is also the GCF of c and c + d.
The GCF of c and d may or may not divide c + d evenly. For example, if c = 2 and d = 4, their GCF is 2, but 2 does not divide 2 + 4 evenly. Therefore, we cannot conclude that the GCF of c and d is also the GCF of c and c + d.
B. 2 + d:
Similarly, the GCF of c and d may or may not divide 2 + d evenly. For example, if c = 3 and d = 5, their GCF is 1, but 1 does not divide 2 + 5 evenly. Therefore, we cannot conclude that the GCF of c and d is also the GCF of c and 2 + d.
Similarly, the GCF of c and d may or may not divide 2 + d evenly. For example, if c = 3 and d = 5, their GCF is 1, but 1 does not divide 2 + 5 evenly. Therefore, we cannot conclude that the GCF of c and d is also the GCF of c and 2 + d.
C. cd:
The product of c and d, cd, will always be divisible by the GCF of c and d. This is because the GCF is a factor of both c and d, and their product will contain the factors of both numbers. Therefore, the GCF of c and d is also the GCF of c and cd.
The product of c and d, cd, will always be divisible by the GCF of c and d. This is because the GCF is a factor of both c and d, and their product will contain the factors of both numbers. Therefore, the GCF of c and d is also the GCF of c and cd.
D. 2d:
The GCF of c and d may or may not divide 2d evenly. For example, if c = 3 and d = 4, their GCF is 1, but 1 does not divide 2 * 4 evenly. Therefore, we cannot conclude that the GCF of c and d is also the GCF of c and 2d.
The GCF of c and d may or may not divide 2d evenly. For example, if c = 3 and d = 4, their GCF is 1, but 1 does not divide 2 * 4 evenly. Therefore, we cannot conclude that the GCF of c and d is also the GCF of c and 2d.
E. d2:
The GCF of c and d may or may not divide d^2 evenly. For example, if c = 2 and d = 3, their GCF is 1, but 1 does not divide 3^2 evenly. Therefore, we cannot conclude that the GCF of c and d is also the GCF of c and d^2.
The GCF of c and d may or may not divide d^2 evenly. For example, if c = 2 and d = 3, their GCF is 1, but 1 does not divide 3^2 evenly. Therefore, we cannot conclude that the GCF of c and d is also the GCF of c and d^2.
By analyzing the options, we can see that the only possible answer is A) c + d. Since we cannot determine that the GCF of c and d is also the GCF of any of the other options, the correct answer is A) c + d.
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If c and d are positive integers and m is the greatest common factor o...
To understand why the correct answer is option 'A', let's break it down step by step:
Given:
- c and d are positive integers
- m is the greatest common factor of c and d
Definition of the Greatest Common Factor (GCF):
The greatest common factor of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder.
Step 1: Understanding the GCF of c and d
Since m is the greatest common factor of c and d, it means that m is the largest positive integer that divides both c and d without leaving a remainder.
Step 2: Considering the options
a) c and d
b) 2 and d
c) cd
d) 2d
Step 3: Analyzing option 'A'
If we choose option 'A' (c and d), we are considering the GCF of c and d itself. Since m is already defined as the GCF of c and d, it follows that m will also be the GCF of c and d. This is because the GCF of any two numbers will always be a factor of those numbers.
Step 4: Analyzing the other options
b) 2 and d: The GCF of 2 and d may or may not be equal to m. It depends on whether or not 2 is a factor of c and d.
c) cd: The GCF of cd is equal to 1, as every positive integer has 1 as a common factor. This is not necessarily equal to m.
d) 2d: The GCF of 2d may or may not be equal to m. It depends on whether or not 2 is a factor of c and d.
Step 5: Conclusion
Based on the analysis above, the only option where m is guaranteed to be the GCF is option 'A' (c and d). Therefore, the correct answer is option 'A'.
Given:
- c and d are positive integers
- m is the greatest common factor of c and d
Definition of the Greatest Common Factor (GCF):
The greatest common factor of two or more integers is the largest positive integer that divides each of the integers without leaving a remainder.
Step 1: Understanding the GCF of c and d
Since m is the greatest common factor of c and d, it means that m is the largest positive integer that divides both c and d without leaving a remainder.
Step 2: Considering the options
a) c and d
b) 2 and d
c) cd
d) 2d
Step 3: Analyzing option 'A'
If we choose option 'A' (c and d), we are considering the GCF of c and d itself. Since m is already defined as the GCF of c and d, it follows that m will also be the GCF of c and d. This is because the GCF of any two numbers will always be a factor of those numbers.
Step 4: Analyzing the other options
b) 2 and d: The GCF of 2 and d may or may not be equal to m. It depends on whether or not 2 is a factor of c and d.
c) cd: The GCF of cd is equal to 1, as every positive integer has 1 as a common factor. This is not necessarily equal to m.
d) 2d: The GCF of 2d may or may not be equal to m. It depends on whether or not 2 is a factor of c and d.
Step 5: Conclusion
Based on the analysis above, the only option where m is guaranteed to be the GCF is option 'A' (c and d). Therefore, the correct answer is option 'A'.
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If c and d are positive integers and m is the greatest common factor of c and d, then m must be the greatest common factor of c and which of the following integers?a)c + db)2 + dc)cdd)2de)d^2Correct answer is option 'A'. Can you explain this answer?
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