Bunuel wrote:
10. The function f is defined for all positive integers a and b by the following rule: f(a,b)=(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?
Notice that the greatest common factor of 10 and x, GCF(10,x), naturally must be a factor of 10: 1, 2, 5, and 10. Thus from f(10,x)=11 we can get four different values of x:
GCF(10,x)=1 --> f(10,x)=11=\frac{10+x}{1} --> x=1;
GCF(10,x)=2 --> f(10,x)=11=\frac{10+x}{2} --> x=12;
GCF(10,x)=5 --> f(10,x)=11=\frac{10+x}{5} --> x=45;
GCF(10,x)=10 --> f(10,x)=11=\frac{10+x}{10} --> x=100.
(1) x is a square of an integer --> x can be 1 or 100. Not sufficient.
(2) The sum of the distinct prime factors of x is a prime number ---> distinct primes of 12 are 2 and 3: 2+3=5=prime, distinct primes of 45 are 3 and 5: 3+5=8\neq{prime} and distinct primes of 100 are 2 and 5: 2+5=7=prime. x can be 12 or 100. Not sufficient.
(1)+(2) x can only be 100. Sufficient.
Answer: C.
Notice that the greatest common factor of 10 and x, GCF(10,x), naturally must be a factor of 10: 1, 2, 5, and 10. Thus from f(10,x)=11 we can get four different values of x:
GCF(10,x)=1 --> f(10,x)=11=\frac{10+x}{1} --> x=1;
GCF(10,x)=2 --> f(10,x)=11=\frac{10+x}{2} --> x=12;
GCF(10,x)=5 --> f(10,x)=11=\frac{10+x}{5} --> x=45;
GCF(10,x)=10 --> f(10,x)=11=\frac{10+x}{10} --> x=100.
(1) x is a square of an integer --> x can be 1 or 100. Not sufficient.
(2) The sum of the distinct prime factors of x is a prime number ---> distinct primes of 12 are 2 and 3: 2+3=5=prime, distinct primes of 45 are 3 and 5: 3+5=8\neq{prime} and distinct primes of 100 are 2 and 5: 2+5=7=prime. x can be 12 or 100. Not sufficient.
(1)+(2) x can only be 100. Sufficient.
Answer: C.
I think the answer to the above questions should be "A" not "C".
Since the GCD for 10 and x can only be 1,2,5 and 10, the corresponding value for x can be 1, 12, 45 and 100. Given this information, the first clause leaves only 100 as the correct answer, which is a square of 10. Please let me know if there is any flaw in my reasoning.