AMC10 2013 10A, Problem 17

Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire.  Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day.  All three friends visited Daphne yesterday.  How many days of the next $365$-day period will exactly two friends visit her?

Solution: From the information above, we get that $A=3x$ $B=4x$ $C=5x$ Now, we want the days in which exactly two of these people meet up The three pairs are $(A,B)$, $(B,C)$, $(A,C)$. Notice that we are trying to find the LCM of each pair. Hence, $LCM(A,B)=12x$, $LCM(B,C)=20x$, $LCM(A,C)=15x$ Notice that we want to eliminate when all these friends meet up. By doing this, we will find the LCM of the three letters. Hence, $LCM(A,B,C)=60x$ Now, we add all of the days up(including overcount). We get $30+18+24=72$. Now, because $60(6)=360$, we have to subtract $6$ days from every pair. Hence, our answer is $72-18=54\implies54$. $\fbox{54}$.

Answer: 54