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21 Jun 2009, 11:31
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(k+1)(k+2)(k+3)
In the above expression, if k is ODD, then (k+1) is EVEN, (k+2) is ODD, & (k+3) is EVEN. The product of all 3 is an EVEN number.
My question: Is (k+1)(k+2)(k+3) divisible by 4? How do you know it is or is not?
In the above expression, if k is ODD, then (k+1) is EVEN, (k+2) is ODD, & (k+3) is EVEN. The product of all 3 is an EVEN number.
My question: Is (k+1)(k+2)(k+3) divisible by 4? How do you know it is or is not?
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21 Jun 2009, 12:04
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topher
You can't be certain it's divisible by 4 - it may be, and it may not be:
If k is odd, then both k+1 and k+3 will be even, and when you multiply two even numbers, you always get a multiple of 4. So if k is odd, then the product will certainly be a multiple of 4.
If k is even, then k+2 is the only even number in the product. If k+2 is divisible by 4, the product will be; if, on the other hand, k+2 is not divisible by 4, then the product will not be divisible by 4. Even numbers alternate between multiples of 4 and non-multiples of 4; if k is divisible by 4, k+2 is not.
Combining the above, we have: The product (k+1)(k+2)(k+3) will be divisible by 4 if and only if k is *not* divisible by 4.
