For two positive integers x and y, we have:
(100*x + y)/(x+y) = 1 + (99*x)/(x+y) and,
(100*x + y)/(x+y) = 100-(99*y)/(x+y) . . . . (i)

Since both x and y are positive, we have: x+y > x and x+y > y

Accordingly, x+y does not divide either x or y, and, consequently from (i) we deduce that x+y divides (100*x + y) whenever, (x+y) divides 99.

Now, substituting (x, y) = (20, 16), we have: x+y = 36 which does not divide 99 and consequently we can conclude that 36 does not divide 2016.

However, in reality 2016/36 = 56 and, accordingly the above result is flawed.

Can you spot the error?

"x+y divides (100*x + y) whenever, (x+y) divides 99" can be rephrased as "if (x+y) divides 99, then x+y divides (100*x + y)".  This is in fact, true.

However, the converse is not true. The error is in assuming that the converse of a true conditional must also be true.

Posted by Charlie on 2012-05-09 13:40:51