1. [ 5 points ] What are the last five digits of the smallest number with exactly 2018 factors (a factor is a positive integer divisor of a number)?


2. [ 5 points ] How many triangles (translations, rotations and reflections are considered different) are there on a 4 by 4 lattice grid of dots, with each dot being a vertex? Two such triangles are shown below:
   


3. The Fibonacci numbers are defined by \[F_1 = 1,F_2=1,F_{n+2}=F_{n+1}+F_n\]
   [ 15 points ] What is the smallest positive integer n such that \(F_n\), the nth fibonacci number, has 2,0,1,8 as the second, third, fourth and fifth digits respectively?
   [ 30 points ] What is the smallest positive integer n such that \(F_n\), the nth fibonacci number, has 2,0,1,8,2,0,1,8 as the second through to nineth digits respectively?


4. n! (n factorial) is defined recursively as \(0!=1\) and \(n!=(n-1)!\times n\) for \(n\geq1\). The number of trailing zeros in a decimal number is the number of zeros with no non-zero digits following it; for example \(13!=6227020800\) has exactly 2 trailing zeros as the last two digits are 0 but the third-last is not. It is possible for n! to have 1 trailing zero; for example, \(7!=5040\) has exactly one trailing zero. It is also possible for n! to have 10, 100, 1000 trailing zeros. However, there exist powers of ten such that n! never has exactly 100...00 trailing zeros.
   [ 10 points ] What is the exponent, k, of the smallest power of ten such that n! never has exactly \(10^k\) trailing zeros?
   [ 35 points ] What is the exponent, k>2018 , of the smallest power of ten greater than \(10^{2018}\) such that n! never has exactly \(10^k\) trailing zeros?