# Problem 350: Constraining the least greatest and the greatest least

![graphic](img350.gif)

A list of size n is a sequence of n natural numbers. Examples are
(2,4,6), (2,6,4), (10,6,15,6), and (11). The greatest common divisor, or
gcd, of a list is the largest natural number that divides all entries of
the list. Examples: gcd(2,6,4) = 2, gcd(10,6,15,6) = 1 and gcd(11) = 11.
The least common multiple, or lcm, of a list is the smallest natural
number divisible by each entry of the list. Examples: lcm(2,6,4) = 12,
lcm(10,6,15,6) = 30 and lcm(11) = 11. Let f(G, L, N) be the number of
lists of size N with gcd ≥ G and lcm ≤ L. For example: f(10, 100, 1) =
91. f(10, 100, 2) = 327. f(10, 100, 3) = 1135. f(10, 100, 1000) mod 1014
= 3286053. Find f(106, 1012, 1018) mod 1014.