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On how many days did they eat the same kind of fruit?

Source: India tst 2002 p9

July 13, 2012
number theorygreatest common divisorinductionalgorithmcombinatorics proposedcombinatorics

Problem Statement

On each day of their tour of the West Indies, Sourav and Srinath have either an apple or an orange for breakfast. Sourav has oranges for the first mm days, apples for the next mm days, followed by oranges for the next mm days, and so on. Srinath has oranges for the first nn days, apples for the next nn days, followed by oranges for the next nn days, and so on. If gcd⁡(m,n)=1\gcd(m,n)=1, and if the tour lasted for mnmn days, on how many days did they eat the same kind of fruit?
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