What's the next number to occur in both sequences?

What's the next number to occur in both sequences?

Question

Consider the arithmetic sequences $1998$, $2005$, $2012$, … and $1996$, $2005$, $2014$, …. What is the next number after $2005$ that appears in both sequences?

Suggestion

What is the common difference for each sequence? How is this useful?

Solution

The sequences have common differences of $7$ and $9$ respectively.

So the numbers in the first sequence can be written as $2005 + 7j$ for whole numbers $j$.

The numbers in the second sequence can be written as $2005 + 9k$ for whole number $k$.

So we are looking for the smallest positive $j$ and $k$ so that $2005 + 7j = 2005 + 9k$, or $7j = 9k$.

Since $7$ and $9$ have no common factor, these values for $j$ and $k$ are $9$ and $7$.

So we have that the next number in both sequences is $2005 + 7.9 = 2068$.

Alternatively, the next number in both lists is $m = 2005 + n$.

Since $m$ is in List $1, 7$ goes into $n$, and since $m$ is in List $2, 9$ goes into $n$.

Thus the smallest possible $n$ is $7 \times 9 = 63$, and $m=2068$ is the number we want.