SOLUTION: 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, b

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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): You can put this solution on YOUR website!

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:



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

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