Written as a fraction in its lowest terms, the recurring decimal 0.123123123… (the block 123 repeating) has which denominator?

The repeating block is three digits long, so multiply by 1000: $1000x-x=123$ gives $999x=123$ and $x=123/999$. Now $123=3\times 41$ while $999=3\times 333$, so the fraction cancels by 3 to 41/333 — and no further, because $333=9\times 37$ shares nothing with the prime 41. The denominator is 333. The trap 999 leaves the fraction uncancelled; 41 reports the numerator instead of the denominator; and 37 is a genuine factor of 333 but not the denominator itself.