jmc

Every positive integer, $n\equiv 23(\mathrm{mod}37),$ can be written in the form: $23+37k$. Thus for every $n<18,632,$ $$0<23+37k<18,632.$$ Since $k$ must be a whole number, $$0\le k\le 502.$$

The set of all $n\equiv 23(\mathrm{mod}37)<18,632$ is then: $$\{23+37(0),23+37(1),23+37(2),...,23+37(502)\}.$$ Counting the number of elements in this set yields $502-0+1=\boxed{503}$ positive integers less than 18,632 which are congruent to $23(\mathrm{mod}37).$