Difference between revisions of "2017 AMC 10B Problems/Problem 14"
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==Solution 1== | ==Solution 1== | ||
| − | By | + | By Fermat's Little Theorem, <math>N^{16} = (N^4)^4 \equiv 1 \text{ (mod 5)}</math> when N is relatively prime to 5. However, this happens with probability <math>\boxed{\textbf{(D) } \frac 45}</math>. |
==Solution 2== | ==Solution 2== | ||
Revision as of 14:02, 30 August 2017
Problem
An integer 
is selected at random in the range 
. What is the probability that the remainder when 
is divided by 
is 
?
Solution 1
By Fermat's Little Theorem, 
when N is relatively prime to 5. However, this happens with probability 
.
Solution 2
Note that the patterns for the units digits repeat, so in a sense we only need to find the patterns for the digits 
.
The pattern for 
is , no matter what power, so doesn't work. Likewise, the pattern for is always . Doing the same for the rest of the digits, we find that the units digits of 
, 
,
, 
,
, 
,
and 
all have the remainder of when divided by , so .
| 2017 AMC 10B (Problems • Answer Key • Resources) | ||
| Preceded by Problem 13 |
Followed by Problem 15 | |
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| All AMC 10 Problems and Solutions | ||
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