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16 Sep 2014, 01:30
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The greatest common divisor of two positive integers is 25. If the sum of the integers is 350, how many such unordered pairs are possible?
A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)
E. \(5\)
A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)
E. \(5\)
16 Sep 2014, 01:30
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Official Solution:
The greatest common divisor of two positive integers is 25. If the sum of the integers is 350, how many such unordered pairs are possible?
A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)
E. \(5\)
We are told that the greatest common factor of two integers is 25. So, these integers are \(25x\) and \(25y\), for some positive integers \(x\) and \(y\). Notice that \(x\) and \(y\) must not share any common factor but 1, because if they do, then the greatest common factor of \(25x\) and \(25y\) will be more than 25.
Next, we know that \(25x+25y=350\). Reducing by 25 gives \(x+y=14\). Now, since \(x\) and \(y\) don't share any common factor but 1, the possible pairs \((x, y)\) can only be (1, 13), (3, 11), or (5, 9) (all other pairs like (2, 12), (4, 10), (6, 8), and (7, 7) share a common factor greater than 1).
So, there are only three pairs of such numbers possible:
\(25*1=25\) and \(25*13=325\);
\(25*3=75\) and \(25*11=275\);
\(25*5=125\) and \(25*9=225\).
Answer: C
The greatest common divisor of two positive integers is 25. If the sum of the integers is 350, how many such unordered pairs are possible?
A. \(1\)
B. \(2\)
C. \(3\)
D. \(4\)
E. \(5\)
We are told that the greatest common factor of two integers is 25. So, these integers are \(25x\) and \(25y\), for some positive integers \(x\) and \(y\). Notice that \(x\) and \(y\) must not share any common factor but 1, because if they do, then the greatest common factor of \(25x\) and \(25y\) will be more than 25.
Next, we know that \(25x+25y=350\). Reducing by 25 gives \(x+y=14\). Now, since \(x\) and \(y\) don't share any common factor but 1, the possible pairs \((x, y)\) can only be (1, 13), (3, 11), or (5, 9) (all other pairs like (2, 12), (4, 10), (6, 8), and (7, 7) share a common factor greater than 1).
So, there are only three pairs of such numbers possible:
\(25*1=25\) and \(25*13=325\);
\(25*3=75\) and \(25*11=275\);
\(25*5=125\) and \(25*9=225\).
Answer: C
General Discussion
