1: Let Rp = 1111 · · · 11, be the positive integer with p digits, all of them 1. For which p is Rp divisible by 37? For which p is Rp divisible by 41?
2: Let Ip,q = 1111 · · · 11 (base q) be the positive integer in base q with p digits, all of them 1. For example, I3,2 is 7 when written in base 10.
(a) Show that Ip,5 is the product of two consecutive integers when p is even.
(b) Show that Ip,9 is a triangle number.
3: We define the eccentricity, E(n), of a positive integer n by the following algorithm. First, E(1) = 1. If the number is odd, then E(n) = 1 + E(n-1/2). If the number is even, E(n) = E(n/2). So, for example, E(11) = 1 + E(5) = 2 + E(2) = 2 + E(1) = 3.
(a) What number(s) less than 2014 have the smallest eccentricity?
(b) What number(s) less than 2014 have the largest eccentricity?
4: Let N be the positive integer with 1998 decimal digits, all of them 1; that is N = 1111 · · · 11.
(a) Determine the sum of the digits of the greatest integer less than √N.
(b) Find the thousandth digit after the decimal point of √N.
5: An upside down LCD?
The familiar LCD display for a calculator uses combinations of 8 line segments to make the digits 0 through 9. What is the thousandth positive integer that is meaningful upside-down (that is which can also be read as a number upside-down)? What is the thousandth positive integer that is upside-down palindromic (that is which reads the same upside-down)? You may assume that the digits 0, 1, 2, 5, 6, 8, 9 read upside down as 0, 1, 2, 5, 9, 8, 6 respectively and that all other digits make no sense read upside down.