BdMO National Junior 2010/3

Discussion on Bangladesh Mathematical Olympiad (BdMO) National
samiul_samin
Posts:1007
Joined:Sat Dec 09, 2017 1:32 pm
BdMO National Junior 2010/3

Unread post by samiul_samin » Sun Feb 24, 2019 9:28 am

One day Tom was playing with numbers.He wrote $11$ fractions using all natural numbers from $1$ to $22$ exactly once-either as numerator or as denomator.How many of these fractions ,at most,are integers?

samiul_samin
Posts:1007
Joined:Sat Dec 09, 2017 1:32 pm

Re: BdMO National Junior 2010/3

Unread post by samiul_samin » Mon Feb 25, 2019 12:09 pm

Answer:$\fbox {10}$

Solution:
a fraction with prime number as denomator and numerator can never be a integer.
Integers are:
\[\dfrac {19}{1}\]

\[\dfrac {14}{2}\]

\[\dfrac {6}{3}\]

\[\dfrac {12}{4}\]

\[\dfrac {15}{5}\]

\[\dfrac {21}{7}\]

\[\dfrac {16}{8}\]

\[\dfrac {18}{9}\]

\[\dfrac {20}{10}\]

\[\dfrac {22}{11}\]

Not integer

\[\dfrac {13}{17}\]

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