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19 Nov 2021, 03:25
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If the sum of two distinct positive integers is 50, then what is the maximum possible value of the greatest common factor of these 2 integers ?
A. \(2\)
B. \(5\)
C. \(10\)
D. \(25\)
E. \(50\)
A. \(2\)
B. \(5\)
C. \(10\)
D. \(25\)
E. \(50\)
19 Nov 2021, 03:25
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Official Solution:
If the sum of two distinct positive integers is 50, then what is the maximum possible value of the greatest common factor of these 2 integers ?
A. \(2\)
B. \(5\)
C. \(10\)
D. \(25\)
E. \(50\)
Test options by starting with largest number.
E. The GCF cannot be 50, because GCF cannot be greater than any one of the numbers, thus cannot equal to their sum.
D. The GCF of 25 would mean that both numbers are 25, which is not the case because we are told that the numbers are distinct.
C. The GCF of 10 would mean that \(x+y=10a+10b=50\), which gives \(a+b=5\). If \((a, b)\) is (1, 4) or (2, 3) (or vise-versa), then the two numbers, \(x\) and \(y\), will be (10, 40) or (20, 30) (or vise-versa). So, GCF of 10 is possible and since its the largest of the possible values among the options, then it must be correct..
Answer: C
If the sum of two distinct positive integers is 50, then what is the maximum possible value of the greatest common factor of these 2 integers ?
A. \(2\)
B. \(5\)
C. \(10\)
D. \(25\)
E. \(50\)
Test options by starting with largest number.
E. The GCF cannot be 50, because GCF cannot be greater than any one of the numbers, thus cannot equal to their sum.
D. The GCF of 25 would mean that both numbers are 25, which is not the case because we are told that the numbers are distinct.
C. The GCF of 10 would mean that \(x+y=10a+10b=50\), which gives \(a+b=5\). If \((a, b)\) is (1, 4) or (2, 3) (or vise-versa), then the two numbers, \(x\) and \(y\), will be (10, 40) or (20, 30) (or vise-versa). So, GCF of 10 is possible and since its the largest of the possible values among the options, then it must be correct..
Answer: C
02 Jan 2023, 04:01
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We can easily eliminate D and E by paying attention to the stem:
It is stated that the two numbers are distinct and positive, so for GCF(x,y)=25, and for x+y=50, x and y must both be 25. Similarly, for GCF(x,y)=50, x+y=50 either x or y must be 0, but they are positive so this is not possible.
We are left with 2,5,10. We can easily make a case for 10, where x=20 and y=30. As this is the largest possible answer choice, this must be correct.
It is stated that the two numbers are distinct and positive, so for GCF(x,y)=25, and for x+y=50, x and y must both be 25. Similarly, for GCF(x,y)=50, x+y=50 either x or y must be 0, but they are positive so this is not possible.
We are left with 2,5,10. We can easily make a case for 10, where x=20 and y=30. As this is the largest possible answer choice, this must be correct.
