1989 AIME Problems/Problem 9
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
Note that is even, since the
consists of two odd and two even numbers. By Fermat's Little Theorem, we know
is congruent to
modulo 5. Hence,
![$3 + 0 + 4 + 7 \equiv n\pmod{5}$](http://latex.artofproblemsolving.com/9/a/e/9ae6c19669910fdcb79f9ce3947ec835f29f0733.png)
![$4 \equiv n\pmod{5}$](http://latex.artofproblemsolving.com/0/1/3/01340d89f298149cc82f56f121d194fd13476f30.png)
Continuing, we examine the equation modulo 3,
![$1 - 1 + 0 + 0 \equiv n\pmod{3}$](http://latex.artofproblemsolving.com/6/2/a/62a72ea43d73fa8703e00716d3a2df730681d17d.png)
![$0 \equiv n\pmod{3}$](http://latex.artofproblemsolving.com/1/2/d/12dfbd3baa056d0fc22f91f6486c78285ec22d0b.png)
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 | |
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All AIME Problems and Solutions |