Difference between revisions of "2007 AIME II Problems/Problem 2"
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== Problem == | == Problem == | ||
− | Find the number of ordered | + | Find the number of ordered triples <math>(a,b,c)</math> where <math>a</math>, <math>b</math>, and <math>c</math> are positive [[integer]]s, <math>a</math> is a [[factor]] of <math>b</math>, <math>a</math> is a factor of <math>c</math>, and <math>a+b+c=100</math>. |
== Solution == | == Solution == |
Revision as of 20:16, 30 July 2009
Problem
Find the number of ordered triples where
,
, and
are positive integers,
is a factor of
,
is a factor of
, and
.
Solution
Denote and
. The last condition reduces to
. Therefore,
is equal to one of the 9 factors of
.
Subtracting the one, we see that . There are exactly
ways to find pairs of
if
. Thus, there are
solutions of
.
Alternatively, note that the sum of the divisors of is
(notice that after distributing, every divisor is accounted for). This evaluates to
. Subtract
for reasons noted above to get
. Finally, this changes
, so we have to add one to account for that. We get
.
See also
2007 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |