2018 AMC 10A Problems/Problem 19
A number is randomly selected from the set
, and a number
is randomly selected from
. What is the probability that
has a units digit of
?
Solution
Since we only care about the unit digit, our set can be turned into
. Call this set
and call
set
. Let's do casework on the element of
that we choose. Since
, any number from
can be paired with
to make
have a units digit of
. Therefore, the probability of this case happening is
since there is a
chance that the number
is selected from
. Let us consider the case where the number
is selected from
. Let's look at the unit digit when we repeatedly multiply the number
by itself:
We see that the unit digit of
for some integer
will only be
when
is a multiple of
. Now, let's count how many numbers in
are divisible by
. This can be done by simply listing:
There are
numbers in
divisible by
out of the
total numbers. Therefore, the probability that
is picked from
and a number divisible by
is picked from
is
.
Similarly, we can look at the repeating units digit for
:
We see that the unit digit of
for some integer
will only be
when
is a multiple of
. This is exactly the same conditions as our last case with
so the probability of this case is also
.
Since
and
ends in
, the units digit of
for some integer
will always be
. Thus, the probability in this case is
.
The last case we need to consider is when the number
is chosen from
. This happens with probability
. We list out the repeading units digit for
as we have done for
and
:
We see that the units digit of
is
when
is an even number. From the
numbers in
, we see that exactly half of them are even. The probability in this case is
Finally, we can ad all of our probabilities together to get
~Nivek