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### Least Number of Marbles

Date: 20 Feb 1995 14:50:47 -0500
From: Anonymous
Subject: I need another marble

Hello,

Can you help with the problem below?

After a successful game of marbles with three friends, Joe said, "If only I
had one more marble, I would have four times as many as Fred, five times
as many as Bill, and seven times as many as John."

What is the least number of marbles Joe could have had?

Date: 20 Feb 1995 23:26:47 -0500
From: Dr. Ken
Subject: Re: I need another marble

Hello there!

We can express the information you gave by the following equation:
let J represent the number of marbles Joe has, and let Fred's, Bill's, and
John's numbers be F, B, and N respectively.  Then we have:
J+1 = 4F = 5B = 7N.

At first glance, it seems like we could just assign any value (like 7 or
something) to one of these variables (like J) and then just solve for the
rest of the variables.  What saves us is the fact that each of J, F, B,
and N must be a non-negative integer.

So let's think about what the equation says.  It says that there is some
number J+1 that's divisible by 4, and by 5, and by 7.  If we're looking for
the smallest number of marbles that Joe could have, then let J+1 be the
smallest number that's divisible by 4, 5, and 7 (namely, 4*5*7=140).  So if
J+1 = 140, J=139.  Right?

I hope this helps you!

-Ken "Dr." Math

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