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 \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>. | + | 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> | ||
Latest revision as of 18:59, 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 
