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March 29, 2015

March 29, 2015

Posted by **zomg** on Thursday, September 14, 2006 at 6:20pm.

Sure

For induction we want to prove some statement P for all the integers. We need:

P(1) to be true (or some base case)

If P(k) => P(k+1) If the statement's truth for some integer k implies the truth for the next integer, then P is true for all the integers.

Look at the first four integers 1,2,3,4. The product 1*2*3*4 is true for the base case n=1,n+1=2,n+2=3 and n+1=4

Suppose now the statement is true for all integers less than or equal to k, so k*(k+1)(k+2)(k+3) is divisible by 24. We want to show that this implies the statement is true for (k+1)(k+2)(k+3)(k+4)

We should observe that with four consecutive numbers 4 will divide one of them say n, and two will divide n-2 or n+2. Three will divide at least one of four consecutive numbers.

You should be able to see that k and k+4 have the same remainder when divided by 4 or 2. This means 8 will divide 4 consecutive numbers. If three does not divide k then it divides one of the next three numbers. If three divides k then it divides k+3. In any case, 8=4*2 and 3 will divide (k+1)(k+2)(k+3)(k+4)

You might be able to simplify my reasoning a little. I didn't proof it closely, so make sure I covered all cases.

thank u very much!

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