Question 733043: Consecutive Integers
If abcd are positive consecutive integers then is a+b^2+c^3 is divisible by d?
If it must always be, prove it. (It does not have to be a formal proof, but you cannot just show me a lot of example.)
If it isnt always, show me an example where it isnt.
Found 2 solutions by lynnlo, solver91311:
Answer by lynnlo(4176)   (Show Source):
Answer by solver91311(24713)  (Show Source):
You can put this solution on YOUR website!
If  ,  ,  , and  are consecutive positive integers, such that  ,  , and  , then
If  is always divisible by then  must have a zero remainder.
Substitute:
^2\ +\ (a\ +\ 2)^3}{a\ +\ 3})
Expand the numerator and collect like terms
Use synthetic division
-3 | 1 7 15 9
| -3 -12 -9
--------------------
1 4 3 0
The remainder is zero, so  where , , , and are consecutive positive integers such that , , and .
John
Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it
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