Find 1 ! + 2 ! + 3 ! + 4 ! + … + 1 0 0 ! ( m o d 1 4 )
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Is there any simpler way than calculating and adding after the remainder is bought down to 1 ! + 2 ! + 3 ! + 4 ! + 5 ! + 6 ! ?
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Well, if you want, you can use the fact that ( n + 1 ) ! = ( n + 1 ) ⋅ n ! in conjunction with modular arithmetic identities so as to ease the calculation of the remainder for the factorials without calculating the factorial itself. Let me give a demonstration.
Take a = 4 and b = 5 . Since, a ! can be computed orally, we find that a ! ≡ ( − 4 ) ( m o d 1 4 ) . Now, you can write b ! = b ⋅ a ! and then use modular arithmetic identities to get 5 ! ≡ 5 ⋅ ( − 4 ) ≡ ( − 2 0 ) ≡ ( − 6 ) ≡ 8 ( m o d 1 4 ) . Checking on a calculator shows that the remainder when 5 ! is divided by 1 4 is 8 indeed. Note that we calculated this without evaluating the factorial. Likewise, we can find the remainder for higher factorials without calculating the factorial itself.
Hope this helps. :)
873 / 14 = 218 with remainder 1 . how it is 5 ???????
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Wow! Sudoku Subbu.Your method of division is great!!.Is 873/14 is 218 with remainder 1 ?
Did you know 14 x 218=3052.Then how can you say that 873/14=218.
@Amrita Roychowdhury , I'll tag you here. I have immense faith that you can post a solution to this. :)
LOL Should I consider deleting this... :P?
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What is wrong with this solution?
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It isn't even a solution -_-
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@Krishna Ar – Relax. Even he should get a chance to impress his girlfriend, shouldn't he?
P.S - This statement was made assuming that the girl he mentioned isn't a relative or a family member. I really hope that my assumption isn't wrong. :3
After 7! There is 14(2*7) in each so they are divisable by 14 so 1!+2!+.....................+7!=873 . When 873 is divided by 14the remainder is 5. So the answer is 5.
why you add 7! .I think the sum should be taken upto 6! as 7! is divisible by 14.
((100!)(100!+1))/2 mod 14 =5
No you can't solve it by that formula because it only holds for the sum of natural numbers...
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We only have to find the sum till 6 ! because the factorials of all the numbers after 6 contain the factor 7 × 2 .Hence all the numbers following 6 ! would be divisible by 14. therefore all we have to find is
1 ! + 2 ! + 3 ! + 4 ! + 5 ! + 6 ! ) ( m o d 1 4 )
= 1 + 2 + 6 + 2 4 + 1 2 0 + 7 2 0 ( m o d 1 4 )
= 8 7 3 ( m o d 1 4 )
= 5