Find the least number b for divisibility What is the smallest positive integer $b$ so that 2014 divides $5991b + 289$?
I just need hints--I am thinking modular arithmetic?
This question was supposed to be solvable in 10 minutes...

As 2014 divides the expression we have 
$$5991b+289=0 \pmod {2014}$$
$$5991b=1725 \pmod { 2014}$$
Now as you said you only want hint you just need to find modulo inverse of 5991 $\pmod{2014}$ and multiply both sides to get $b$. ( Note as 5991 and 2014 are co-prime the inverse modulo of 5991 $\pmod{2014}$ exists). Also If you are new to modular arithmetic then look up modulo inverse, when it exists and extended euclid gcd method to find it. Hope it helps.
