Moving Statue
Find the coefficient of x 5 0 in the expansion ( x − 1 ) ( 2 x − 1 ) ( 3 x − 1 ) ⋯ ( 5 0 x − 1 ) .
Submit your answer as the last non-zero digit of the number you've found.
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2 solutions
That is a trivial observation. But could you elaborate on how one can find the last non-zero digit of 5 0 ! ?
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Some most have used google
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Maybe. Although, there actually is a simple way to do this using modular arithmetic. See the solutions and comments in the solutions discussion of this similar problem to get an idea about how to calculate the last non-zero digit of n ! for non-negative integer n .
The basic algorithm is to "clear out" the trailing zeros of n ! by dividing it with ν 5 ( n ! ) , where ν 5 ( n ! ) denotes the 5 -adic order of n ! for non-negative n . Then, you compute modulo 1 0 of this "reduced" value to get the answer to the original problem. You'll also need to use Chinese Remainder Theorem, Extended Euclidean Algorithm and Wilson's Theorem to do the modular calculations properly.
is there a term of x^50 ?
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There is .
Note that it is a degree 5 0 polynomial that we are given and as such, we will have 5 1 terms in the expansion where one of the terms will be independent of x .
The answer should be 8
and how would u evaluate the last non zero digit of 50! @Tanishq Varshney
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50! Has 12 trailing 0s. Hence finding:
1 0 1 2 5 0 ! modulo 10 would work. This is done by multilplying out all the units digits of 50! using modulo 10, leaving out all the 5s and 12 factors of 2 in the product.
I'm too lazy to do this right now, so I got the answer wrong :p
Take − 1 common in all term and multiplying we see a 50th degree polynomial comes up whose last term is 5 0 ! x 5 0
Coefficient of x 5 0 = 5 0 !