A friend who teaches middle-school math recently asked me whether it was a coincidence that every positive integer and its fifth power seem to have the same last (i.e., rightmost) decimal digit. I said that it was definitely not a coincidence and that it followed from a theorem in number theory called Euler’s theorem, which I explained briefly. Right after this, however, I recalled that since the theorem applies only to numbers that are relatively prime to the modulus (which is 10 in this case, since dividing any positive integer by 10 and taking the remainder gives us that integer’s last digit), the theorem applies only to numbers ending in 1, 3, 7, or 9, and cannot be applied to numbers ending in 0, 2, 4, 5, 6, or 8, since they are not relatively prime to 10. Nonetheless, we can easily check that in this case, his observation works for all these numbers as well:
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The author wishes to thank Keith Conrad, Alice Dean, Joe Silverman, and the reviewer for helpful comments on this paper. The author also wishes to thank Gary Mullen for the ongoing mathematical inspiration he is willing to share.
K. Conrad, Euler’s Theorem, available at math.uconn.edu/~kconrad/blurbs/ugradnumthy/eulerthm.pdf.Google Scholar
J. H. Silverman. A Friendly Introduction to Number Theory, fourth edition. Pearson Prentice Hall, 2013.Google Scholar