Math Chat: Powers of 5 and Traveling To the North Pole

Old challenge (Howard Sheldon)

What is the remainder when you divide 5999,000 by 7?


The remainder is 1, as deduced by several readers who recognized that the remainder of powers of 5 after dividing by 7 repeats in cycles of 6.

When you ignore multiples of 7 and look only at the remainder, you are doing arithmetic "modulo 7," as if 7 = 0, 25 = 4 (both are 4 more than multiples of 7), and 699 = -1. Modulo 7, 52 = 25 = 4, 53 = 4 x 5 = 20 = -1, and 56 = 53 x 53 = (-1) x (-1) = 1; since 999,000 = 6 x 166,500, therefore 5999,000= (56)166,500 = 1166,500 = 1.

It is no accident that powers modulo 7 repeat in cycles of 6. For any prime number p, powers repeat in cycles of p minus 1. This important fact in number theory is called Fermat's Little Theorem, not to be confused with the famous 350-year-old Fermat's Last Theorem, which got much attention when it was finally proved recently by Andrew Wiles.

This week we would like to credit the new readers who solved the challenge problem: E. Bright, T. Chang, G. Chapman, R. Daniels, E. Dravecky, M. Hildebrand, M. Howard, S. Longworth, E. Randolph, and G. Rice.

Harder challenge

We sometimes use the symbol ^ to denote powers, so that 5^999,000 denotes 5x5x...x5 (999,000 times), 5^5 = 55 = 3125, and 5^5^5 denotes 5^(5^5) = 53125 (not 31255). William Foster asks for the remainder after dividing by 7 of 5^5^ ... ^5 (999,000 times).

Calendar puzzles (Edward Wallner, from the British Games & Puzzles magazine No. 60, May 1977)

Miguel de Cervantes and William Shakespeare both died on April 23, 1616. Who died first?

The English archbishop Whitgift, first chairman of the committee that produced the authorized version of the Bible, died on Feb. 29, 1603. Explain how there could be a Feb. 29 in a year not divisible by four.


Cervantes of Spain, author of Don Quixote, died first. As a Roman Catholic country, Spain immediately implemented the Gregorian calendar's 10- day advance of 1582. Shakespeare's Protestant England did not implement the reform until 1752. So in 1616, April 23 occurred 10 days earlier for Cervantes than for Shakespeare.

In England in 1603, the year started on March 25, in the so-called Annunciation style. Therefore the day after March 24, 1603, was March 25, 1604. A leap day still occurred on Feb. 29 as though the year had started on Jan. 1.


This spring's advance to daylight saving time in most of the United States prompted an interesting Kidspace quiz on time on the Home Forum Page April 1. Unfortunately, it had some errors in the first two answers. The mean solar day is indeed a bit longer than the period of Earth's rotation ("sidereal day"), because in a day Earth has also advanced about 1/365th of the way around the sun and therefore must turn about an extra 1/365th day, or four minutes, to face the sun again. Contrary to the given answer, however, it is the mean solar day that is almost exactly 24 hours (as everyone with a clock knows), while the sidereal day is about four minutes shorter.

As for the number of time zones spanned by the US, there are actually only seven from the Virgin Islands to the Aleutian Islands, and only six if you stick to the States. (If you count the Virgin Islands, you should also count other territories such as Guam.)

New Northwest challenge (Eric Brahinsky)

Suppose you start at the earth's equator and travel continuously northwest until you reach the North Pole.

What does your path look like? How long is your path?

* To be eligible for 'Flatland' and other book awards, send answers and new questions to:

Math Chat, Bronfman Science Center, Williams College Williamstown, MA 01267

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