If a and b are positive integers, what is the greatest possible value of a?
Though you might be tempted to run for your calculator, this problem can be solved using a couple of logic-based approaches. Indeed, no intense calculation needed, just some application of exponents.
, which approximates to 500. So , or slightly less than 100. Therefore, the answer is most likely (D). However, you might argue, 500 isn’t quite 512, and that little bit of difference could make a whole lot of difference when taken to the eleventh power.
Notice the answers, though. There is no twelve, but only thirteen. So if it is a hard call, without a calculator, to know if that extra 500 will make the left side greater than the right, we don’t need to worry: 12 is not an answer. (E) 13, which would add two ‘500s’ to the left, or about would definitely go over . In other words, that slight difference between 500 and 512, taken to the eleventh power, would be much smaller than . Therefore, answer (D).