Question 203853
The key to the consecutive integer (or consecutive even or odd integer) problems is the answer to the question: "How much more is each of these integers from the one before it?"<br>
For consecutive integers the answer is: 1
Then, to write expressions for consecutive integers, make the variable represent the lowest integer and then, for the rest of the integers, keep adding 1:
<pre>
Lowest:       x
Next higher:  x + 1
Next higher:  x + 1 + 1 = x + 2
Next higher:  x + 2 + 1 = x + 3
Next higher:  x + 3 + 1 = x + 4
Next higher:  x + 4 + 1 = x + 5
etc.
</pre>
For consecutive even (or odd) integers the answer to the key question is: 2.
Then, to write expressions for consecutive even (or odd) integers, make the variable represent the lowest integer and then, for the rest of the integers, keep adding 2:
<pre>
Lowest:       x
Next higher:  x + 2
Next higher:  x + 2 + 2 = x + 4
Next higher:  x + 4 + 2 = x + 6
Next higher:  x + 6 + 2 = x + 8
Next higher:  x + 8 + 2 = x + 10
etc.
</pre>
In your problem you want 4 consecutive odd integers so you will use x, x+2, x+4 and x+6. It says their sum is 117. Sum means addition so:
{{{x + (x+2) + (x+4) + (x+6) = 117}}}
Now we just solve this equation. Start by simplifying:
{{{4x + 12 = 117}}}
Subtract 12 from both sides:
{{{4x = 105}}}
Divide by 4:
{{{x = 105/4 = 26.25}}}
Since this is not an integer, much less an odd integer, it means the problem, as stated is impossible. There are no 4 odd consecutive integers which add up to 117.