Difference between revisions of "2021 JMPSC Sprint Problems/Problem 8"

(Solution)
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<cmath>10, 20, 30, 40, 50, 60, 70, 80, 90</cmath>
 
<cmath>10, 20, 30, 40, 50, 60, 70, 80, 90</cmath>
  
Therefore, there are only <math>9</math> that satisfy the requirements.
+
Therefore, there are only <math>9</math> two-digit numbers that do not satisfy the requirements. There are <math>100-11+1=90</math> two-digit numbers total, so there are <math>90-9=\boxed{81}</math> numbers.
  
 
-OofPirate
 
-OofPirate

Revision as of 11:27, 11 July 2021

Problem

How many positive two-digit numbers exist such that the product of its digits is not zero?

Solution

Rather than counting all the two-digit numbers that exist with those characteristics, we should do complementary counting to find the numbers with the product of its digits as 0.

The only numbers with $0$'s in their digits are the multiples of $10$.

\[10, 20, 30, 40, 50, 60, 70, 80, 90\]

Therefore, there are only $9$ two-digit numbers that do not satisfy the requirements. There are $100-11+1=90$ two-digit numbers total, so there are $90-9=\boxed{81}$ numbers.

-OofPirate