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

(Created page with "==Problem== How many positive two-digit numbers exist such that the product of its digits is not zero? ==Solution== asdf")
 
 
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==Solution==
 
==Solution==
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asdf
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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.
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The only numbers with <math>0</math>'s in their digits are the multiples of <math>10</math>.
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<cmath>10, 20, 30, 40, 50, 60, 70, 80, 90</cmath>
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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.
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-OofPirate
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== Solution 2 ==
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You don't want a digit in this number to contain <math>0</math>. Therefore, the answer is <math>9 \cdot 9=\boxed{81}</math>
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- kante314 -
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==See also==
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#[[2021 JMPSC Sprint Problems|Other 2021 JMPSC Sprint Problems]]
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#[[2021 JMPSC Sprint Answer Key|2021 JMPSC Sprint Answer Key]]
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#[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]]
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{{JMPSC Notice}}

Latest revision as of 09:44, 12 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

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

The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition. JMPSC.png

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