rahulpadhy's blog

By rahulpadhy, history, 8 years ago, In English

Can anyone tell me how to approach this question ? I have been trying it for quite sometime now, but to no avail.. The link to the question is : http://www.spoj.com/problems/EASYFACT/

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8 years ago, # |
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Any sum of consecutive integers can be described by a low value a and a length r. So we must have:

Try simplifying this expression and then factoring it.

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    8 years ago, # ^ |
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    Can you please explain this one a bit more ? I had got this expression, but unable to proceed any further..

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      8 years ago, # ^ |
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      Sure. To get that first formula, notice that

      Simplifying the formula in my previous comment, we get

      2N! = r(r + 2a + 1).

      Instead of counting all pairs (r,a) we will count all pairs (r,r+2a+1) because it's simpler: now we're just counting all ordered pairs (x,y) with xy = 2N! where x<y and x and y have different parity.

      This is equivalent to counting the number of odd divisors of 2N!. I can explain the solution to this reduced problem if you're still stuck.

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        8 years ago, # ^ |
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        Suppose that I consider n = 5. Now if I am counting the number of odd divisors of (2 * 5 !), then I am getting the odd divisors as (1, 3, 5, 15), which is 4. But in the question, we can see that the answer for n = 5 is 2, i.e., when we are considering the sum of 3 consecutive positive integers(39 + 40 + 41) and the sum of 5 consecutive positive integers(22 + 23 + 24 + 25 + 26). So still stuck..

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          8 years ago, # ^ |
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          Since we have to remove the case when r=1, we must subtract 1 from the final answer (so for n=5, we get 3). The answer for n=5 should be 3, since we also have the sum of 15 consecutive positive integers (120 = 1 + 2 + ... + 14 + 15).

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        8 years ago, # ^ |
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        .

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        2 years ago, # ^ |
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        This may be sound silly but how did you concluded that "This is equivalent to counting odd divisors of 2N!"