Suppose that a and b are integers with the property
that ab + 1 is a multiple of 24.

Show that a + b is also a multiple of 24.

Let ab + 1 = 24k for some k. Then 24k - ab = 1.

If a has a common factor m with 24, then m would divide evenly into
the left hand side, and hence m would divide evenly into 1.
Hence the only common factor of a and 24 is 1, and the same
argument applies to b as well.

As a result, when a or b are divided by 24, the only possible remainders
are 1, 5, 7, 11, 13, 17, 19 and 23.

And since ab + 1 is a multiple of 24, it's easy to check that the only
possible combinations of remainders for a and b are
(1 and 23), (5 and 19), (7 and 17), and (11 and 13).

For each of these cases, it is clear that a + b will be a multiple
of 24.