2010 AMC 12A Problems/Problem 20
Problem 20
Arithmetic sequences and
have integer terms with
and
for some
. What is the largest possible value of
?
Solution
Since and
have integer terms with
, we can write the terms of each sequence as
where and
are the common differences of each, respectively.
Since
it is easy to see that
.
Hence, we have to find the largest such that
and
are both integers.
The prime factorization of is
. We list out all the possible pairs that have a product of
and soon find that the largest value is
for the pair
, and so the largest
value is
.