^{5}and n end in the same digit for all positive integers n?

This is an obvious generalization of Problem E32.

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A12. Fifth powers final digit - generalized.

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E32. Fifth powers final digit

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If numbers are expressed in base b, for which b is it true that n^{5} and n end in the same digit for all positive integers n?

This is an obvious generalization of Problem E32.

This is an obvious generalization of Problem E32.

The following rather neat problem occurs in *Challenging Problems in Algebra* by Alfred S. Posamentier and Charles T. Salkind. I think it is suitable for a bright high school student or, with some hints, even for average high school students.

Prove that n and n^{5} always end in the same digit (in ordinary base-10 representation).

Prove that n and n

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From time to time I will post an elementary mathematics problem which I hope readers may enjoy. "Elementary" to me means that the problem does not require any specialized mathematical knowledge beyond high-school mathematics to solve. Some of these elementary problems will be very simple, others will require a great deal of cleverness. Elementary math problems will be denoted by the letter E followed by the problem number.

I will sometimes post a link to a solution. If I haven't yet posted a link, you may send me your answer and if it is correct I will credit you on the blog. To send answers, please mailto: peterash3@gmail.com.

Thanks,

Peter

I will sometimes post a link to a solution. If I haven't yet posted a link, you may send me your answer and if it is correct I will credit you on the blog. To send answers, please mailto: peterash3@gmail.com.

Thanks,

Peter

From time to time post problems that are somewhat more advanced than those in the Elementary Math Problems. These problems will require a knowledge of some college-level mathematics, either for their statement or for the solution that I know. Advanced math problems will be denoted by the letter A followed by the problem number.

I will sometimes post a link to a solution. If I haven't yet posted a link, you may send me your answer and if it is correct I will credit you on the blog. To send answers, please mailto: peterash3@gmail.com.

I will sometimes post a link to a solution. If I haven't yet posted a link, you may send me your answer and if it is correct I will credit you on the blog. To send answers, please mailto: peterash3@gmail.com.

Here are some recent reviews on mathematics, learning theory, education, and related technology:

The Number Sense: How the Mind Creates Mathematics by Stanislaus Dehaene, Oxford University Press 1997

Three Books on the Riemann Hypothesis

The King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts, Walker and Company, 2006

The Poincare Conjecture: In Search of the Shape of the Universe, Donal O'Shea, Walker & Company, 2007

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, George Lakoff and Rafael E. Nunez, Basic Books, 2000.

The Number Sense: How the Mind Creates Mathematics by Stanislaus Dehaene, Oxford University Press 1997

Three Books on the Riemann Hypothesis

The King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts, Walker and Company, 2006

The Poincare Conjecture: In Search of the Shape of the Universe, Donal O'Shea, Walker & Company, 2007

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, George Lakoff and Rafael E. Nunez, Basic Books, 2000.

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