2014 AMC 12B Problems/Problem 7
Problem
For how many positive integers is also a positive integer?
$\textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}}\ 7\qquad\textbf{(E)}\ 8$ (Error compiling LaTeX. Unknown error_msg)
Solutions
Solution 1
We know that or else will be negative, resulting in a negative fraction. We also know that or else the fraction's denominator will exceed its numerator making the fraction unable to equal a positive integer value. Substituting all values from to gives us integer values for . Counting them up, we have possible values for .
Solution 2
Let , where . Solving for , we find that . Because and are relatively prime, . Our answer is the number of proper divisors of , which is .
See also
2014 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 6 |
Followed by Problem 8 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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