Problem:
One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed that there is a positive integer n such that
1335+1105+845+275=n5.
Find the value of n.
Solution:
It is clear that n≥134. We can get an upper bound on n by noting that
n5=1335+1105+845+275<1335+1105+(27+84)5<3(133)5<10243125(133)5=(45)5(133)5
Thus n<(45)(133), giving n≤166. Next note that, when an integer is raised to the fifth power, its units digit is unchanged. It follows that n has the same units digit as the sum 133+110+84+27; i.e., the units digit of n is 4, and n is one of the four numbers 134,144,154,164. Since 133≡1(mod3),110≡2(mod3), 84≡0(mod3) and 27≡0(mod3), we have
n5=1335+1105+845+275≡15+25≡0(mod3)
This means that n is a multiple of 3, and we conclude that n=144.
The problems on this page are the property of the MAA's American Mathematics Competitions