Difference between revisions of "1989 AIME Problems/Problem 9"
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== Problem == | == Problem == | ||
| − | One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that <math>133^5+110^5+84^5+27^5=n^{5 | + | One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that <math>133^5+110^5+84^5+27^5=n^{5}</math>. Find the value of <math>n</math>. |
== Solution == | == Solution == | ||
Revision as of 21:10, 17 July 2008
Problem
One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that 
. Find the value of 
.
Solution
By Fermat's Little Theorem, we know 
is congruent to modulo 5. Hence,
Continuing, we examine the equation modulo 3,
Thus, is divisible by three and leaves a remainder of four when divided by 5. It's obvious that 
, so the only possibilities are 
or 
. It quickly becomes apparent that 174 is much too large, so must be 
.
See also
| 1989 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 8 |
Followed by Problem 10 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
