Consecutive Odd Integers

Date: 08/28/97 at 18:32:11
From: Heather Mahurin
Subject: Addition-Word Problem

I need to know three consecutive odd integers that equal 150.

Date: 08/28/97 at 19:45:13
From: Doctor Barney
Subject: Re: Addition-Word Problem

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
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