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 $2^n \nequiv 0$ (Error compiling LaTeX. Unknown error_msg) 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