1998 CEMC Gauss (Grade 7) Problems/Problem 25
Problem
Two natural numbers, and
do not end in zero. The product of any pair,
and
is a power of 10 (that is, 10, 100, 1000, 10 000 , ...). If
, the last digit of
cannot be
Solution
If the product is a power of
and both
and
do not end in 0, then
must be in the form
and
must be in the form
We know that for all positive integers
and
for all integers
.
Start looking at small values of
and subtract:
This pattern continues in groups of , and the only number not included is
-edited by coolmath34