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### Same Day Off?

```Date: 12/08/2002 at 22:41:27
From: Tabitha Mole
Subject: Math word problems

One person has every ninth day off and started on Sunday, and
another person has every sixth day off and started on Monday. When
will they both be off on the same day?

I have tried to use a calendar to count out the days and could not
get it to work out.

I don't think they will ever be off on the same day.
```

```
Date: 12/08/2002 at 23:17:54
From: Doctor Ian
Subject: Re: Math word problems

Hi Tabitha,

Let's see how things start out:

S M T W T F S | S M T W T F S | S M T W T F S

Person 1  w w w w w w w   w - w w w w w   w w w - w w w

Person 1    w w w w w -   w w w w w - w   w w w w - w w

So it looks as if the first person has days

9, 18, 27, ..., 9*n, ...

off, and the second person has off

7, 13, 19, ..., (1 + 6*m), ...

So the question is: What are the smallest integer values of n and m
for which

9n = 6m + 1

n = (6m + 1)/9

Now, in order for n to be an integer, (6m + 1) must be a multiple of
9.

But let's think about that for a minute.  The number 6m will _always_
be a multiple of 3, right? So the number (6m + 1) will _never_ be a
multiple of 3, which means that it will never be a multiple of 9,
either.

So you're right - they never get the same day off!

- Doctor Ian, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
Middle School Algebra