2023 AMC 10B Problems/Problem 8
Contents
Problem
What is the units digit of ?
Solution 1
(mod 10). ~andliu766
The method omits many steps, and it is important to note that we do not have . For example, , but .
Solution 2
When looking at the units digit patterns of the powers of , we see that
, units digit
, units digit
, units digit
, units digit
, units digit
And the pattern repeats. This pattern will apply for the powers of as well, since the units digit of is . We can find the pattern for the powers of too. The pattern follows with units digits, , , , , , , ... Similarly, the units digit of will follow the same pattern as the powers of .
Both of these powers cycle in groups of . When diving by , we get remainder , meaning complete cycles; or the power being a multiple of , times, and extra. So the units digit of is . divided by is reminder , which means complete cycles, or the power being a multiple of , times, and extra. So the units digit of is .
We only need to find the units digit in the end, so we just add those already found units digits, to get a new units digit of . Therefore the answer is
~mk00
Solution 3 (Digit Cycles)
Note that the units digit will be the same regardless of the tens, hundreds, and thousands digits, so we can simplify this problem to finding the last digit of . We can find the units digit of , by listing the units digits of the first few powers of two, and trying to find a pattern.
As we can see the units digits of powers of two repeat after every four iterations. Now we know the units digit of is and the units digit of . Similarly we can find the last digits of powers of three repeat after every four, so the units digit of is . Adding these together, the ones digit is the same as the ones digit of which is .
~vsinghminhas
Solution 4 (Bashy)
~ Yrock
See also
2023 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
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All AMC 10 Problems and Solutions |
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