Question

The highest power of 2 in 10! + 11!+12!+13!+---+1000! is?

Answer 1

how many 2 can you factor out in 10!?

Since the GCD of (10!, 11!, ..., 1000!) is 10!, we just have to consider 10!.

Answer 2

\[\large\begin{array}{rcl}
10!&=&1\times2\times3\times4\times5\times6\times7\times8\times9\times10\\
&=&2\times3\times2^2\times5\times2\times3\times7\times2^3\times3^2\times2\times5\\
&=&2\times2^2\times2\times2^3\times2\times3\times5\times3\times7\times3^2\times5\\
&=&2^{1+2+1+3+1}\times3^{1+1+2}\times5^{1+1}\times7\\
&=&2^8\times3^4\times5^2\times7
\end{array}\]

Answer 3

Therefore the answer is 8

(only if i have interpreted your question correctly)

Answer 4

Yes, what exactly does the question mean?

Answer 5

continuing kc trick :)


(10! + 11!+12!+13!+---+1000! )

10!(1+11 + 11x12 + 11x12x13 + .... )
10!(12 + 11x12 + 11x12x13 + ...)
10!*12(1 + 11 + 11x13 + 11x13x14+...)
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since gcd(12, 11x13) = 1, we cannot factor out anymore.

so we need to consider 10!*12

Answer 6

wow @ganeshie8 i completely missed out 12 xDDDDDDD