Difference between revisions of "1956 AHSME Problems/Problem 48"
(Created page with "== Problem 48== If <math>p</math> is a positive integer, then <math>\frac {3p + 25}{2p - 5}</math> can be a positive integer, if and only if <math>p</math> is: <math> \tex...") |
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<cmath>\frac{3p+25}{2p-5} = 1 + \frac{p+30}{2p-5}</cmath> | <cmath>\frac{3p+25}{2p-5} = 1 + \frac{p+30}{2p-5}</cmath> | ||
− | Therefore, in order for <math>\frac{3p+25}{2p-5}</math> to be a positive integer, <math>\frac{p+30}{2p-5}</math> must be a non-negative integer. Since the bottom the the fraction is an odd number, we can multiply the top of <math>\frac{p+30}{2p-5}</math> by 2 without changing whether it is an integer or not. Therefore, in order for <math>\frac{3p+25}{2p-5}</math> to be an integer, <math>\frac{2p+60}{2p-5} = 1 + \frac{65}{2p-5}</math> must also be an integer. As a result, <math>2p-5</math> must be a factor of 65, or <math>p = | + | Therefore, in order for <math>\frac{3p+25}{2p-5}</math> to be a positive integer, <math>\frac{p+30}{2p-5}</math> must be a non-negative integer. Since the bottom the the fraction is an odd number, we can multiply the top of <math>\frac{p+30}{2p-5}</math> by 2 without changing whether it is an integer or not. Therefore, in order for <math>\frac{3p+25}{2p-5}</math> to be an integer, <math>\frac{2p+60}{2p-5} = 1 + \frac{65}{2p-5}</math> must also be an integer. As a result, <math>2p-5</math> must be a factor of 65, or <math>p = 3, 5, 9, 35</math>. Therefore <math>p</math> must be at least 3, and less than or equal to 35. So the answer which best fits these constraints is <math>\boxed{\textbf{(B)}}</math>. |
Latest revision as of 05:05, 13 January 2024
Problem 48
If is a positive integer, then
can be a positive integer, if and only if
is:
Solution
Lets begin by noticing that:
Therefore, in order for to be a positive integer,
must be a non-negative integer. Since the bottom the the fraction is an odd number, we can multiply the top of
by 2 without changing whether it is an integer or not. Therefore, in order for
to be an integer,
must also be an integer. As a result,
must be a factor of 65, or
. Therefore
must be at least 3, and less than or equal to 35. So the answer which best fits these constraints is
.