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1. What is the smallest positive integer \(n\) such that the rightmost three digits of \(n!\) and \((n+1)!\) are the same?

2. For how many positive integers \(n\) less than 100 is \(5^n+8^{n+1}+13^{n+2} \) a multiple of 6?

3. A group of monsters landed on Earth and started reproducing by each splitting into two every hour. For any positive integer \(k\) we use \(N_k \) to denote the number of monsters \(k\) hours after landing. We know that \(N_0<5\) (fewer than 5 monsters landed) and that the two digits at the right end of \(N_{100}\) are \(28\). How many monsters landed on Earth at the beginning?

 Oct 3, 2022
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1 - The smallest positive integer n ==10

10! =3,628,800

(10 + 1)! =11! ==39,916,800

 

 

2 - There is a mistake in this question. Check your numbers carefully.

 

3 - N_100 =2^100 =1,267,650,600,228,229,401,496,703,205,376

N_k =3 - Number of monsters

N_k  *  N_100 =3 x  1,267,650,600,228,229,401,496,703,205,376=3,802,951,800,684,688,204,490,109,616,128

 Oct 4, 2022

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