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?
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
