The Discreet Charm of the DS
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05 Feb 2012, 07:11
10. The function \(f\) is defined for all positive integers \(a\) and \(b\) by the following rule: \(f(a,b)=\frac{(a+b)}{GCF(a,b)}\), where \(GCF(a,b)\) is the greatest common factor of \(a\) and \(b\). If \(f(10,x)=11\), what is the value of \(x\)?
Observe that the greatest common factor of 10 and \(x\), \(GCF(10,x)\), must be a factor of 10. Hence, \(GCF(10,x)\) can be 1, 2, 5, or 10. Thus, from \(f(10,x)=11\), we can derive the following four different values of \(x\):
• If \(GCF(10,x)=1\), then \(f(10,x)=11=\frac{10+x}{1}\), and \(x=1\);
• If \(GCF(10,x)=2\), then \(f(10,x)=11=\frac{10+x}{2}\), and \(x=12\);
• If \(GCF(10,x)=5\), then \(f(10,x)=11=\frac{10+x}{5}\), and \(x=45\);
• If \(GCF(10,x)=10\), then \(f(10,x)=11=\frac{10+x}{10}\), and \(x=100\).
(1) \(x\) is the square of an integer.
This information tells us that \(x\) can be 1 or 100. Not sufficient.
(2) The sum of the distinct prime factors of \(x\) is a prime number.
The distinct prime factors of 12 are 2 and 3: \(2+3=5=prime\), the distinct prime factors of 45 are 3 and 5: \(3+5=8 \ne prime\), and the distinct prime factors of 100 are 2 and 5: \(2+5=7=prime\). Therefore, \(x\) can be 12 or 100. Not sufficient.
(1)+(2) \(x\) can only be 100. Sufficient.
Answer: C