1961 AHSME Problems/Problem 28
Contents
[hide]Problem 28
If is multiplied out, the units' digit in the final product is:
Solution
has a unit digit of . has a unit digit of . has a unit digit of . has a unit digit of . has a unit digit of .
Notice that the unit digit eventually cycles to itself when the exponent is increased by . It also does not matter what the other digits are in the base because the units digit is found by multiplying by only the units digit. Since leaves a remainder of after being divided by , the units digit of is , which is answer choice .
Alternate Solution
- ( ): If is a prime and is an integer prime to then we have .
- Let's define () as units digit funtion of .
We can clearly observe that,
()=
. .
. .
. .
(1$$ (Error compiling LaTeX. Unknown error_msg)
and we can see by Fermat's Theorem that this cycle repeats with the cyclicity of . Now = + 7)7 ~GEOMETRY-WIZARD $
See Also
1961 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 27 |
Followed by Problem 29 | |
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