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Omni-MATH / The classrooms at MIT are each identified with a positive integer (with no…

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problem

The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let NN be the original number of the room, and let MM be the room number as shown on the sign. The smallest interval containing all possible values of MN\frac{M}{N} can be expressed as [ab,cd)\left[\frac{a}{b}, \frac{c}{d}\right) where a,b,c,da, b, c, d are positive integers with gcd(a,b)=gcd(c,d)=1\operatorname{gcd}(a, b)=\operatorname{gcd}(c, d)=1. Compute 1000a+100b+10c+d1000 a+100 b+10 c+d.
Plain-text mathematical notation (without MathML)
The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let N be the original number of the room, and let M be the room number as shown on the sign. The smallest interval containing all possible values of (M)/(N) can be expressed as [(a)/(b),(c)/(d)) where a,b,c,d are positive integers with gcd(a,b)=gcd(c,d)=1. Compute 1000a+100b+10c+d.
Original LaTeX notation
The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let $N$ be the original number of the room, and let $M$ be the room number as shown on the sign. The smallest interval containing all possible values of $\frac{M}{N}$ can be expressed as $\left[\frac{a}{b}, \frac{c}{d}\right)$ where $a, b, c, d$ are positive integers with $\operatorname{gcd}(a, b)=\operatorname{gcd}(c, d)=1$. Compute $1000 a+100 b+10 c+d$.

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Source and history

task_id
f54077d5-fa45-5ebc-965b-a135696b5a52
task_key
test--f54077d5-fa45-5ebc-965b-a135696b5a52
task_revision_id
1
config / split
/ test

Official source

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