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

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.