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Machine Pizza

Date: 08/15/97 at 07:05:06
From: windows
Subject: Pizza machine


A machine for processing pizzas puts salami on top of every 18th 
pizza, adds pieces of pineapple to every 21st pizza, and adds peppers 
every 20th pizza. Since the machine started working, it has processed 
100000 pizzas.

  1. How many peppered pizzas with salami and pineapple has the 
     machine processed?

  2. How many peppered pizzas with salami but without pineapple has 
     the machine processed?

Thank you!

Date: 08/20/97 at 16:07:37
From: Doctor Rob
Subject: Re: Pizza machine

I will assume that the first pizza with salami is the 18th, the first
with pineapple is the 21st, and the first with peppers is the 20th.
The actual answer will change if this is not so.  Another easily
envisioned start is if all three ingredients are added to the first
pizza.  The conditions do not say what starting scenario is used out
of the 7560 possibilities.

  1. [100000/L], where L is the least common multiple of 18, 21, 
     and 20. This is because every L pizzas, all three ingredients 
     are added.

  2. [100000/18] - [100000/M], where M is the least common multiple 
     of 18 and 21.  This is because every 18 pizzas, salami is added, 
     but every M pizzas, both salami and pineapple are added.

-Doctor Rob,  The Math Forum
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