Decals on Lockers

Date: 02/28/2002 at 21:39:18
From: Shemla
Subject: Word problem

Dear Dr Math,
 
Can you please help me with this math problem?

The lockers in Pythagoras Middle School are numbered 1 through 500. 
Starting the count with locker 1, every 6th locker has a blue decal, 
evey 9th locker has a yellow decal, and every 10th locker has a green 
decal. How many lockers have all three decals?

Thank you.

Date: 03/01/2002 at 14:24:34
From: Doctor Sarah
Subject: Re: Word problem

Hi Shemla - thanks for writing to Dr. Math.

Let's start by drawing a chart. *'s will be lockers, B is for a blue 
decal, and Y is for a yellow decal.

                 18                  36                  54
*****B**** *B*****B** ***B*****B *****B**** *B*****B** ***B**** *B****

********Y* *******Y** ******Y*** *****Y**** ****Y***** ***Y**** ****Y*

We can see that there's a repeating pattern: at every third B decal 
there's also a Y decal on the same locker. These pairs occur at 
lockers 18, 36, 54... and these are numbers divisible by both 6 and 9.

What numbers less than 500 are divisible by 6, 9, and 10? (Look for 
the least common multiple of those three numbers, and then multiples 
of that number less than 500.)

Can you take it from here?

- Doctor Sarah, The Math Forum
  http://mathforum.org/dr.math/