Matrix Puzzle - Math Puzzle

Source: www.puzzletweeter.com

Problem:
Let A,B be 2x2 matrices with integer entries.
Suppose the matrices A,A+B,A+2B,A+3B,A+4B are all invertible and their inverses are also integer matrices.

Then show that A+5B is invertible and it's inverse is an integer matrix.

Update (24 June 2014):
Solution: Posted by Yashoteja Prabhu (Ex-RA at Microsoft Research, IITB CSE 2011 Alumnus) and Sanchit Gupta in comments!

Art of Puzzle Solving

Art of Puzzle Solving from Pratik Poddar

Equations Puzzle

Source: Interview Street

Problem: 

Find the number of positive integral solutions for the equations 

(1/x) + (1/y) = 1/N! 

(read 1 by x plus 1 by y is equal to 1 by n factorial) 

Print a single integer which is the no of positive integral solutions modulo 1000007.

Update (24 June 2014):
Solution: Posted by JDGM (James D G Miles) in comments!