Why does that one (the 4096) question need a knowledge of powers of two? You just need powers of one, if I read it correctly.
Let j and k be positive integers. What is the largest possible value of j+k given that j^2 = k^2 + 252?
I’d say that a good start would be to remember that it is the same as j+k * j-k = 252 and that 16^2 = 256. 
On that 4096 question I almost allowed hubris to lead me into the trap of thinking about powers of 2 exclusively. I was very proud of myself that I could quickly do powers of 2 up to 4096 in my head and immediately recognized the question as a prime favtoring question and was ready to congratulate myself on a job well done when I, just for the sake of completeness, considered if there was any other way we could get x and y to be further apart. Alas 4096^1! I am assuming the answer is 4095. That is some nasty stuff!
Actually now that I am looking back at it x had to be the base and y the exponent so even if the base were 2 the answer would have been negative and would have been less negative of the base were 4 or something larger than 2 so even for that reason x should not have been 2. Just goes to show you what happens when you do before you think instead of the other way around!
The question that started this thread: “Blue Book, page 919 prob. 16. If xy=7 and x-y=5, then x^2y-xy^2 = ??? a. 2, b. 12, c. 24, d 35, e. 70…”
shows up ALL THE TIME. Do enough SAT prep, and you’ll see it over and over again.
Basic rule for SAT prep: if it can be factored, factor it. If it can be multiplied out, multiply it out.
It’s like the problem of the number of poles needed to hold up a 300 foot wire if there’s a pole every 50 feet. You have to remember to include the last pole. And it shows up all the time!
Do enough SAT prep, and you’ll recognize the questions that show up on a regular basis.
@xiggi ah, but (j, k) = (16, 2) is not the optimal solution.
(This question is probably equivalent to level 5, but the solution is pretty short.)
I am feeling pretty humbled. That question that you posted @MITer94…
Let j and k be positive integers. What is the largest possible value of j+k given that j^2 = k^2 + 252?
I like your starting point Xiggi. But from there I didn’t use the 16^2 = 256 part. I did something pretty different and came up with 126. I am not sure if that is right, but if it is that felt pretty hard. And if it isn’t then I guess it is even harder than it felt, lol!!! Can I assume that is not an SAT question? If so I was completely missing something and again maybe jumping into things before thinking it through.
I feel like this thread has gotten pretty derailed!!! I am enjoying it nonetheless!
@reasonsat yes it’s 126. It was just a question I made up, but I feel like this could be a suitable level 5 question (or maybe a tad harder since most “plugging in” strategies probably don’t work).
A simple algebraic solution is to use difference of squares as mentioned before:
(j-k)*(j+k) = 252
We want to maximize j+k so we minimize j-k (note that they’re both positive ints since j, k are positive ints).
Trying 1*252 = 252, this doesn’t give us a solution because 1 and 252 are of different parity. A simple way to see that there is no solution is to solve j-k = 1, j+k = 252 --> j = 253/2.
Trying 2*126 = 252, since 2 and 126 have the same parity, we obtain a solution where j+k = 126, so 126 is optimal (j=64, k=62).
Ultimately I did what you did and realized that j+k and j-k both had to be even so after 1 and 252 I tried 2 and 126, as you suggested. But I did several other things along the way, including prime factoring 252, which led nowhere. After using difference of the squares I also tried to think of it as j plus some quantity (k) and j minus that same quantity quantity (k) both had to be factors of 252. So 18 and 14 are paired factors of 252 and so it could be 16 plus 2 and 16 minus 2. But the obviously we want j+k to be as big as possible so we want to look for factors that are spaced much farther apart. Looking back I could have pursued that line of reasoning and got to it but in the end I abandoned it and essentially did what you suggested above. I knew that j+k and j-k had to be factors of 252 so I tried them spaced far apart (1 and 252) and then solved for j and k and realized that they j+k and j-k had to be even for j and k to end up being integers. So then I tried 2 and 126 and then solved for j and k.
But that was pretty hard! I think that is a little beyond what you would see on the SAT. In my opinion people tend to really struggle with number property questions, even pretty mild ones, so they don’t have to make those kinds of questions that hard to stump most people on the test. I think that would stump almost all test takers, certainly given the time constraints.
MIT, fwiw, I did not want to type the optimal solution in case others wanted to play. The 16^2 was to show where a more “square” form of j+k and j-K might lead to and be the limit. A more “rectangular” form (if that makes visual sense) was to lead to an optimal shape with a maximum value for j+k.
My paper looked like this
252 1
126 2 ✓
84 3
63 4
42 6
36 7
18 14
Harder to explain than to do! My geometrical explanation might not make immediate sense. 
But I should add that I liked the question! Very saucy!
@xiggi Ah, I see.
@reasonsat thanks!