2021 JMPSC Sprint Problems/Problem 8

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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

Solution 2

You don't want a digit in this number to contain $0$. Therefore, the answer is $9 \cdot 9=\boxed{81}$

- kante314 -

See also

  1. Other 2021 JMPSC Sprint Problems
  2. 2021 JMPSC Sprint Answer Key
  3. All JMPSC Problems and Solutions

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