The four-digit number $52\times 6$ is divisible by 12. What is the digit $x$?

Divisibility by 12 needs BOTH tests to pass. For 4, the last two digits $x$6 must be a multiple of 4, which allows $x=1$, 3, 5, 7 or 9. For 3, the digit sum $5+2+x+6=13+x$ must be a multiple of 3, which allows $x=2$, 5 or 8. Only $x=5$ survives both, and $5256=12\times 438$. The distractors each pass exactly one test: 3 passes only the 4-test, while 2 and 8 pass only the 3-test.