Difference between revisions of "2013 Mock AIME I Problems/Problem 2"
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== Solution == | == Solution == | ||
− | <math>\boxed{005}</math>. | + | Because <math>a|b</math>, let <math>b=an</math>, where <math>n</math> is a positive integer. Because <math>c-a=10</math>, <math>c=a+10</math>, so <math>(b+1)|(a+10)</math> and thus <math>(an+1)|(a+10)</math>. Now, let <math>a+10=m(an+1)</math>, where <math>m</math> is another positive integer. Thus, <math>a=\tfrac{10-m}{mn-1}</math>. Because the ordered pair <math>(m,n)</math> uniquely determines values of <math>a</math>, <math>b</math>, and <math>c</math>, the desired number of triples <math>(a,b,c)</math> that fit the constaints of the problem equals the number of positive integer pairs <math>(m,n)</math> that force <math>a=\tfrac{10-m}{mn-1}</math> and, consequently, <math>b</math> and <math>c</math>, to be positive integers. |
+ | Starting with <math>n=1</math>, by listing out fractions of the form <math>\tfrac{10-m}{mn-1}</math> and seeing if they simplify to positive integers, we see that the only possible values of <math>m</math> are <math>2</math> and <math>4</math>. Likewise, for <math>n=2</math>, <math>m</math> must be <math>1</math>. For <math>n=4</math>, <math>m=1</math>, and for <math>n=10</math>, <math>m=1</math>. No other values of <math>n</math> yield positive integer values of <math>m</math>. Thus, because there are <math>5</math> ordered pairs <math>(m,n)</math>, our answer is <math>\boxed{005}</math>. | ||
==See also== | ==See also== |
Latest revision as of 11:50, 30 July 2024
Problem
Find the number of ordered positive integer triplets such that
evenly divides
,
evenly divides
, and
.
Solution
Because , let
, where
is a positive integer. Because
,
, so
and thus
. Now, let
, where
is another positive integer. Thus,
. Because the ordered pair
uniquely determines values of
,
, and
, the desired number of triples
that fit the constaints of the problem equals the number of positive integer pairs
that force
and, consequently,
and
, to be positive integers.
Starting with
, by listing out fractions of the form
and seeing if they simplify to positive integers, we see that the only possible values of
are
and
. Likewise, for
,
must be
. For
,
, and for
,
. No other values of
yield positive integer values of
. Thus, because there are
ordered pairs
, our answer is
.