Summation problem 2015
x = 1 ∑ 2 0 ( x 6 + 2 0 1 5 )
Using only paper and a writing instrument, what is the greatest prime factor of the expression above?
The answer is 7216537.
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1 solution
Okay, people could reach to the answer that x = 1 ∑ 2 0 ( x 6 + 2 0 1 5 ) = 2 1 6 4 9 6 1 1 0
Now 2 1 6 4 9 6 1 1 0 = 2 × 3 × 5 × 7 2 1 6 5 3 7
But how can anybody, just using paper and writing instrument, come to know that this big 7216537 is a prime number? For that they'll have to check if any 1 out of all the primes till 2686 divides the number, does that sound simple? (Clicked "reviewed" for the problem though...)
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I had to google it out
There's an easy way to check for divisibility by 3 and 7.
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And also it's advisable to use the formula for consecutive sums for the 6th power to expedite the process
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@Hobart Pao – And of course, make good use of factoring skills. I made and solved this entire problem by hand.
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@Hobart Pao – You're not getting my point! The thing is, after you get the factors 2,3,5 and the big number 7 2 1 6 5 3 7 , how do you decide that the big number is a prime? You just can't use the divisibility tests there my friend, because you'll have to check divisibility for all primes till the number 2 6 8 6 . That can't be done by hand >_<
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@Aditya Raut – You only need to make about 800 calculations (assuming you know what are the primes from 2 to 2686)... Or :p Pascal, maybe.
Thanks for the link.
I'm too lazy to write a full solution, but basically, split the two sums so that you have x = 1 ∑ 2 0 x 6 + ( 2 0 1 5 ⋅ 2 0 ) . There's a formula for the sum of x = 1 ∑ 2 0 x 6 , you can find this online at the following link: http://www.math.rutgers.edu/~erowland/sumsofpowers.html
Then this becomes a factoring problem, which I'm sure you all are capable of.