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 This brainteaser was written by Derrick Niederman. The number groups below are the last five digits of the fifth powers of the numbers 31 through 39. However, the groups aren’t in the right order to represent the fifth powers of 31 through 39 sequentially. Using only these digits, and without using a calculator, can you place the groups in the correct order? 35393 35424 29151 24199 21875 35168 54432 43957 66176 Resources for Teaching Math
© 2009 National Council of Teachers of Mathematics, Inc.
http://illuminations.nctm.org
Solution: 29,151; 54,432; 35,393; 35,424; 21,875; 66,176; 43,957; 35,168; 24,199. If you look for a pattern, you’ll find that the units digit of the fifth power of any integer is the same as the units digit of the original number. This fact is obviously true if the units digit is 0, 1, or 5. But is it true for the other integers 2–9? Let’s start by looking at numbers with units digit 2. You probably know that 22 = 4, 23 = 8, 24 = 16, and 25 = 32, and the units digit of the fifth power is 2. Try another number with units digit 2, let’s say 12. Then 122 = 144, 123 = 1728, 124 = 20,736, and 125 = 248,832. Again, the units digit of the fifth power is 2. And if you try it for other numbers that have units digit 2, you’ll find the same thing, though the numbers start to get fairly large. More generally, if the units digit of n is 2, then the units digit of n2 is 4. (This may seem obvious to you, but if not, there’s a simple proof. Any number with units digit 2 can be written in the form 10x + 2. When this number is squared, the result is (10x + 2)2 = 100x2 + 40x + 4. That result is 4 more than a multiple of 10, so its units digit must be 4.) Likewise, the units digit of n3 is 8. (Again, look at (10x + 2)3 = 1000x3 + 600x2 + 120x + 8, which is 8 more than a multiple of 10, so its units digit must be 8.) Continuing, the units digit of n4 is 6, and the units digit of n5 is 2. A similar result will be found if you examine numbers with units digit 3, 4, 6, 7, 8, and 9. In every instance, the units digit of the fifth power of any integer is the same as the units digit of the original number. Therefore, the units digit of 305 is 0, the units digit of 315 is 1, the units digit of 325 is 2, and so forth. This gives the answer, and the set of last five digits just need to be arranged so that the units digits are in order from 0 to 9: 29,151 54,432 35,393 35,424 21,875 66,176 43,957 35,168 24,199 In an advanced number theory course, you might learn that xp – 1 = 1 mod p, for any integer x and prime number p. This implies that x4 = 1 mod 5, and multiplying both sides by x gives x5 = x mod 5. It’s also the case that x5 = x mod 2, which is just a fancy way of saying that the fifth power of an odd number is odd and the fifth power of an even number is even. Combining these two facts allows us to conclude that x5 = x mod 10. (That’s because if a = b mod 5, then either a = b mod 10 or a = b + 5 mod 10, but if a = b mod 2, then the latter possibility is ruled out.) This is the fact we need to know for this problem and proves that the units digit of the fifth power is equal to the units digit of the original number. Resources for Teaching Math
© 2009 National Council of Teachers of Mathematics, Inc.
http://illuminations.nctm.org