What a coincidence!
Consider two arithmetic sequences: 1998, 2005, 2012,... and 1996, 2005, 2014,... Which numbers will appear in both?
Age
11 to 14
Challenge level
Problem
Consider the arithmetic sequences:
$1998, 2005, 2012, ...$ and
$1996, 2005, 2014, ...$
$1996, 2005, 2014, ...$
Which is the next number after $2005$ that appears in both sequences?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
The sequences have common differences of $7$ and $9$ respectively.
The lowest common multiple of $7$ and $9$ is $63$, so the next term after $2005$ to appear in both sequences is $2005 + 63$, that is $2068$.