## Constraining the least greatest and the greatest least

### Problem 350

Published on Saturday, 10th September 2011, 07:00 pm; Solved by 284; Difficulty rating: 60%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 101^{4} = 3286053.

Find f(10^{6}, 10^{12}, 10^{18}) mod 101^{4}.