A large wholesale company for books uses three different types of shelf in their ware- houses. Their capacity is gauged in terms of a certain specimen book of average size, known under the nickname "Dumbbook". Shelves of type A can accommodate 56 Dumbbooks, the ones of type B only 45 Dumbbooks, while the extra large type C shelves hold 121 Dumbbooks. Upon receiving a certain batch of books (assumed to be Dumbbooks), the workers first put them into shelves of type A, filling a certain number of shelves exactly and leaving 11 books left over. They then try to t the books into shelves of type B, filling a certain number of shelves exactly and leaving space for 23 books on the last shelf. Finally they try to t the books into shelves of type C, filling a certain number of shelves exactly and leaving 4 books left over. What is the number of books, assuming all batches have at most 10000 books?
I have summarised this into the system of congruences:
x = 11 mod 56
x = 23 mod 45
x = 4 mod 121
and I get the solution to be 183803 mod 304920, which I have checked and seems correct. However, this is clearly below 10000, and since gcd(56,45,121) = 1, I believe this means there is a unique solution. Does this mean that there are no appropriate solutions? Or have I done something wrong?
What you did wrong is in reading the statements.
"... and leaving space for 23 books on the last shelf" means the second equation (for shelf B) should read $$ x = -23 \mod 45 $$ Redo your work with this different second equation and I think you will find that the solution comes out to be less that 10,000.