Try this beautiful problem from Algebra: Counting Days

## Counting Days – AMC-10A, 2013- 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?

,

- $48$
- $54$
- $60$
- $65$
- \(72\)

**Key Concepts**

Algebra

LCM

## Check the Answer

Answer: $54$

AMC-10A (2013) Problem 17

Pre College Mathematics

## Try with Hints

Given that Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day.

According to the questation , Let us assume that $A=3 x B=4 x C=5 x$

Now, we want the days in which exactly two of these people meet up

The three pairs are $(A, B),(B, C),(A, C)$

Can yoiu find out the LCM of each pair……….

Can you now finish the problem ……….

\(LCM(A, B)=12 x, LCM(B, C)=20 x, LCM(A, C)=15 x\)

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, $LCM(A, B, C)=60 x$

can you finish the problem……..

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 $72-18=54$

## Other useful links

- https://www.cheenta.com/surface-area-of-cube-amc-10a-2007-problem-21/
- https://www.youtube.com/watch?v=VLyrlx2DWdA&t=20s