What is the greatest n = ? n =?

Algebra Level 3

If 1 0 n 10^n divides ( 2 10 5 ) ( 5 2 10 ) 1 0 5 2 \Large\frac{(2^{10{^ 5}})(5^{2^{10}})}{10^{5^{2}}} ,

then what is the greatest value of n = ? n=?


The answer is 999.

Hana Wehbi by Hana Wehbi , 51, USA

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1 solution

Naren Bhandari
Jul 28, 2018

D = 2 1 0 5 ⋅ 5 2 10 1 0 5 2 = ( 2 1 0 5 − 25 ) ⋅ ( 5 2 10 − 25 ) = ( 1 0 2 10 − 25 ) ⋅ 2 1 0 5 − 25 − ( 2 10 − 25 ) D = \dfrac{2^{10^5} \cdot 5^{2^{10}}}{10^{5^2}} = \left(2^{10^5 -25} \right)\cdot \left(5^{2^{10 }- 25}\right) =\left(10^{2^{10}-25}\right)\cdot 2^{10^5-25-\,(2^{10}-25)} Here, 1 0 2 10 − 25 = 1 0 999 10^{2^{10}-25} = 10^{999} , Thus largest possible value of n = 999 n=\boxed{999}

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