K and L are each four-digit positive integers with thousands, hundreds, tens, and units digits defined as a, b, c, and d, respectively, for the number K, and p, q, r, and s, respectively, for the number L. For numbers K and L, the function W is defined as $\frac{5^a2^b7^c3^d}{5^p2^q7^r3^s}$. The function Z is defined as $\frac{K-L}{10}$. If $W = 16$, what is the value of Z?

Wait, so W is basically the ratio of the prime factorizations of K and L? That's a wild setup. I get that W=16=2^4, but I don't immediately see how the digits connect to K-L. Can someone nudge me on why we care about 10 in the denominator of Z?

This one took me a while. The trick is realizing W compares the two numbers' digit powers, not the numbers themselves. Once I wrote K and L in expanded form the rest clicked. Solid 700-level question.

K and L are both four-digit, so a and p are nonzero. Does that constraint matter here, or is it just there to keep the numbers legit?

Got 40 and felt good about it until I second-guessed the last step. Is it just me or is the 'cannot be determined' choice a trap for people who don't pin down the digits?