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### Consecutive Odd Integers

```
Date: 08/28/97 at 18:32:11
From: Heather Mahurin

I need to know three consecutive odd integers that equal 150.
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```
Date: 08/28/97 at 19:45:13
From: Doctor Barney

First, we should approach this problem with the understanding that
we have no guarantees that there are three consecutive odd integers
that add up to 150, but let us assume that there are, and in this
manner we may either find three of them or prove that there are no
such three numbers.

Let us name these three unidentified integers X, Y, and Z, for
convenience, until we find their real values.  Since they are
consecutive odd integers, we know that the largest is 2 more than
the middle one, and that the middle one is 2 more than the smallest.

Arbitrarilly assuming that X is the smallest and Z is the largest,
we may write: X+2 = Y and Y+2 = Z.

We also know that X+Y+Z = 150. Now I am going to eliminate variables
from this equation by "substitution". If Z = Y+2, that means that
Z is the same number as Y+2, whatever numbers Z and Y really are,
so anywhere there is a Z I can put in a Y+2 instead without changing
the truth of the equality.

X+Y+(Y+2) = 150.  Now I will do the same thing using Y = X+2.

X+(X+2)+((X+2)+2) = 150   Next add up all the X's and all the 2's

3X+6 = 150   Now subtract 6 from both sides of the
equation
3X = 144   Finally solve for X (have you had algebra
yet?) by dividing by 3
X = 48

But X is even!  This means that there are not any three consecutive
odd integers that add up to 150. Actually, it is fairly easy to prove
that any three odd integers can never add up to an even number: any
two odd integers will add up to an even number, and adding a third odd
number will always create an odd number for the final sum.

Can you find three consecutive even integers that add up to 150?

-Doctor Barney,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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