Problem of the Month
Problem 0: September 2022

1. Consider the integers \(392\), \(487\), \(638\), and \(791\). For each of these integers, do the following.

   1. Determine whether the integer is a multiple of \(7\).

   2. With the hundreds digit equal to \(A\), the tens digit equal to \(B\), and the units digit equal to \(C\), compute \(2A+3B+C\).

   What do you notice?

2. Suppose \(n=ABC\) is a three-digit integer (\(A\) is the hundreds digit, \(B\) is the tens digit, and \(C\) is the units digit). Show that if \(ABC\) is a multiple of \(7\), then \(2A+3B+C\) is a multiple of \(7\).

3. Show that if \(2A+3B+C\) is a multiple of \(7\), then the three-digit integer \(n=ABC\) is a multiple of \(7\).

4. Suppose \(ABCDEF\) is a six-digit integer that has each of its digits different from \(0\). Show that \(ABCDEF\) is a multiple of \(7\) if and only if \(BCDEFA\) is a multiple of \(7\).

5. Think of ways to generalize the fact in part (d).