Difference between revisions of "1997 AIME Problems/Problem 3"
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[[Category:Intermediate Number Theory Problems]] | [[Category:Intermediate Number Theory Problems]] | ||
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Revision as of 18:34, 4 July 2013
Contents
Problem
Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?
Solution
Let 
be the two-digit number, 
be the three-digit number. Putting together the given, we have 
. Using SFFT, this factorizes to 
, and 
.
Since 
, we can use trial and error on factors of 1000. If 
, we get a non-integer. If 
, we get 
and 
, which satisifies the conditions. Hence the answer is 
.
Solution 2
As shown above, we have 
, so 
. 
must be just a little bit smaller than 9, so we find , , and the solution is 
.
See also
| 1997 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
