## Linear Combinations of Semiprimes

### Problem 278

Published on Saturday, 13th February 2010, 05:00 am; Solved by 560

Given the values of integers 1 a1 a2 ... an, consider the linear combination
q1a1 + q2a2 + ... + qnan = b, using only integer values qk 0.

Note that for a given set of ak, it may be that not all values of b are possible.
For instance, if a1 = 5 and a2 = 7, there are no q1 0 and q2 0 such that b could be
1, 2, 3, 4, 6, 8, 9, 11, 13, 16, 18 or 23.
In fact, 23 is the largest impossible value of b for a1 = 5 and a2 = 7.
We therefore call f(5, 7) = 23.
Similarly, it can be shown that f(6, 10, 15)=29 and f(14, 22, 77) = 195.

Find f(p*q,p*r,q*r), where p, q and r are prime numbers and p < q r 5000.