1989 AIME Problems/Problem 9
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 .
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 .
We can cheat a little bit and approximate, since we are dealing with such large numbers. As above, , and it is easy to see that . Therefore, , so the last digit of is 4.
We notice that and are all very close or equal to multiples of 27. We can rewrite as approximately equal to . This means must be close to .
134 will obviously be too small, so we try 144. . Bashing through the division, we find that , which is very close to . It is clear that 154 will not give any closer of an answer, given the rate that fifth powers grow, so we can safely assume that is the answer.
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