2017 AMC 10B Problems/Problem 14
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
.
Solution 3 (casework)
We can use modular arithmetic for each case of
If , then
If , then
If , then
If , then
If , then
In out of the
cases, the result was
, and since each case occurs equally, the answer is
2017 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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