This brainteaser was supplied by the Mathematical Olympiads for Elementary and Middle Schools (www.moems.org). 7772 means 777 × 777, 7773 means 777 × 777 × 777, and so on. Suppose 7777 is completely multiplied out. What is the units digit of the resulting product? Resources for Teaching Math
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Solution: 3. If you multiply out 7772, you get 777 × 777 = 603,729. The units digit is 9. If you multiply out 7773, you get 777 × 777 × 777 = 469,097,433. The units digit is 3. You could continue to multiply things out, but the numbers start to get very large in a hurry. Instead, it might be better to notice that the units digit of 72 = 49 is the same as the units digit of 7772, and the units digit of 73 = 343 is the same as the units digit of 7773. This might lead you to speculate that the units digit of 77 is the same as the units digit of 7777, and you’d be correct. Note that 77 = 823,453, so the units digit of 77 and 7777 is 3. As it turns out, the units digit of a product is only influenced by the units digits of the factors. So, it doesn’t actually matter that the number 777 is being multiplied. If we used the number 1,234,567 instead, we’d find that the units digit of 1,234,5677 is also 3, because the units digit of 1,234,567 is 7. In fact, the units digit for the seventh power of any number with units digit 7 will be 3. Resources for Teaching Math
© 2009 National Council of Teachers of Mathematics, Inc.
http://illuminations.nctm.org