Question

find the integer value of a and b such that 30a -41b =1​

Answer 1

More direct path to a+b+c=12:

Write x=a+b+c. In particular the last equation implies 31x≥365 so x>11. On the other hand, the first equation is 28x+2b+3c=365. If either b or c is nonzero, this means 28x<364 so x<13. And b=c=0 is not possible because 365 is not divisible by 28.

added: to be fair I should complete the proof... x=12 means 2b+3c=29. c must be odd and less than 10, so it can be 1,3,5,7,9. Substitute in 2b+3c=29, then in a+b+c=12 and keep only the solutions with non-negative a.