2022 AMC 12B Problems/Problem 15
Problem: One of the following numbers is not divisible by any prime number less than 10. Which is it?
Solution 1 (Process of Elimination)
We examine option E first. has a units digit of
(Taking the units digit of the first few powers of two gives a pattern of
) and
has a units digit of
(Taking the units digit of the first few powers of three gives a pattern of
). Adding
and
together, we get
, which is a multiple of
, meaning that
is divisible by 5.
Next, we examine option D. We take the first few powers of added with
:
We see that the odd powers of added with 1 are multiples of three. If we continue this pattern,
will be divisible by
. (The reason why this pattern works: When you multiply
by
, you obtain
. Multiplying by
again, we get
. We see that in every cycle of two powers of
, it goes from
to
and back to
.)
Next, we examine option B. We see that has a units of digits of
(Taking the units digit of the first few powers of two gives a pattern of
). Adding
to
, we get
. Since
has a units digit of
, it is divisible by
.
Lastly, we examine option A. Using the difference of cubes factorization , we have
. Since
(Every term in the sequence is equivalent to
),
is divisible by
.
Since we have eliminated every option except C, is not divisible by any prime less than
.