jmc

Observing that neither $957$ nor $1537$ can be divided by $2,$ $3,$ $5,$ or $7,$ we turn to the Euclidean algorithm. We get $$\begin{array}{rl}\text{gcd}(957,1537)& =\text{gcd}(957,1537-957)\\ & =\text{gcd}(957,580)\\ & =\text{gcd}(580,957-580)\\ & =\text{gcd}(580,377)\\ & =\text{gcd}(377,580-377)\\ & =\text{gcd}(377,203)\\ & =\text{gcd}(203,174)\\ & =\text{gcd}(174,203-174)\\ & =\text{gcd}(174,29)\\ & =\boxed{29}.\end{array}$$Remark: Note that we could have computed $\text{gcd}(957,580)$ more quickly by observing that $$\text{957 is divisible by neither 2 nor 5}$$$$⟹\text{gcd}(957,580)=\text{gcd}(957,58).$$The greatest common divisor of 957 and 58 can be calculated using the Euclidean algorithm in one step: long division gives $957÷58=16\text{ remainder }29$.