Newspaper article The Christian Science Monitor

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

Newspaper article The Christian Science Monitor

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

Article excerpt

Old challenge (Howard Sheldon)

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

Answer 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 5999,000 denotes 5x5x...x5 (999,000 times), 55 = 55 = 3125, and 555 denotes 5(55) = 53125 (not 31255). William Foster asks for the remainder after dividing by 7 of 55 ... 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. Answer Cervantes of Spain, author of Don Quixote, died first. …

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