Difference between revisions of "1998 CEMC Gauss (Grade 7) Problems/Problem 25"
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If the product <math>pq</math> is a power of <math>10,</math> and both <math>p</math> and <math>q</math> do not end in 0, then <math>p</math> must be in the form <math>5^n</math> and <math>q</math> must be in the form <math>2^n.</math> | If the product <math>pq</math> is a power of <math>10,</math> and both <math>p</math> and <math>q</math> do not end in 0, then <math>p</math> must be in the form <math>5^n</math> and <math>q</math> must be in the form <math>2^n.</math> | ||
− | We know that <math>5^n \equiv 5 (\ | + | We know that <math>5^n \equiv 5 (\pmod 10)</math> for all positive integers <math>n</math> and <math>2^n \not\equiv 0 (\pmod 10)</math> for all integers <math>n</math>. |
Therefore, we know that <math>p - q \not\equiv 5 - 0 = \boxed{\textbf{(C)}\ 5}.</math> | Therefore, we know that <math>p - q \not\equiv 5 - 0 = \boxed{\textbf{(C)}\ 5}.</math> |
Revision as of 17:58, 7 April 2024
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
.
Therefore, we know that