Hey guys,
This may be a day late and a dollar short (it looks like the thread itself is about 8 months old), but I saw this near the top of the list and thought I'd chime in. Regarding number properties, I'd say that the best way to learn them is to identify them and apply them yourself. A few key points:
1) When the GMAT is testing large (i.e. 12^34) or abstract numbers (i.e. n^(x+1) - n^(x-1)) there's a good chance that number properties are in play. In these cases, see if you can establish a pattern. For example, I worked with a student on a question that included 2^x - 2^(x-2), and set that equal to a large number (3*2^17). Knowing that the question dealt with abstract and large numbers, I sought to find a pattern using small numbers: 2^3 - 2^1 = 6, which is 3(2^1). 2^4 - 2^2 = 12, which is 3(2^2), etc. By noting that there was a pattern - the difference is always 3 * the second term - I could easily solve this one.
2) It is more important to know that number properties exist than to memorize the property itself. By definition, these properties will always hold, so if you need to remember them, you can always prove them. For example, the number 7, when taken to exponents 1, 2, ...., n goes through the cycle: 7, 9, 3, 1, 7, 9, 3, 1, etc. But you don't need to memorize that pattern - as long as you know that "exponents yield pattern-driven units digits", you can rather quickly prove the pattern for each number to yourself, and save your mental energy when you study for other things.
3) Most importantly, number properties is more a state of mind than a list of rules. When you solve problems, see if you can find patterns between them, or use number properties to find a faster, smarter way to do it. The more you learn to think in those terms, the more you'll have a feel for number properties as a whole. The nice thing for the GMAT is that "number properties" includes a variety of individual numerical rules, so they can build the reward structure to reward higher-order thinking and not just memorization; the nice thing for you is that, once you buy in to the mindset, even the most complex math problems on the test are solvable by finding that pattern or mode of thinking that unlocks them.
Brian Galvin
GMAT Instructor
Chief Academic Officer
Veritas Prep
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