Problem Solving

tdot_gmat-er wrote:Hey guys, so here's the first question I saw on the GMAT Prep test. Needless to say, I am stumped. Any ideas?

If (1/5)^m * (1/4)^18 = 1 / (2 * (10)^35), what's the value of m?

Also, what's the best way to approach number theory problems? I've been through OG, Kaplan, and PR but they really don't provide much strategy for these. Do you guys have any strategies or resources?

Thx!

about the problem:

first off, notice that the fractions are meant as a distraction: all of their numerators are 1, so the problem is really about the denominators. once you realize that, just rewrite the problem for the denominators only:
(5^m)(4^18) = (2)(10^35)

once you've seen one problem like this, you've seen them all: the key is to break everything down into primes. the problem appears to be difficult right now, because certain numbers 'overlap' each other (i.e., there are 2's contained in the 4's and 10's); once you break the numbers down into primes, there's no such 'overlap' anymore because prime factorizations are unique.

therefore:
(5^m)(4^18) = (2)(10^35)
(5^m)((2^2)^18) = (2)((2*5)^35)
(5^m)(2^36) = (2)(2^35)(5^35)
(5^m)(2^36) = (2^36)(5^35)
5^m = 5^35
m = 35

if your understanding of primes is deep enough, you may have noticed from the start that you can actually ignore all the 2's in the problem: since all that's missing is a power of 5, the 2's are guaranteed to cancel.

make sure that you have a very thorough grasp of the laws of exponents, as they are vital to a great many gmat math problems. a good grip on all the rules of algebra 1 is a good thing, of course, but the laws of exponents are perhaps even more useful than the rest of the lot.

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resources:

i noticed that you haven't mentioned the manhattangmat number properties guide in your list. sad face :(

seriously though, our guide has some good stuff in it, and is a nice springboard for a general understanding of number properties.

another strategy is to study the o.g. solutions for ALL problems having anything to do with number properties - whether you got them right or missed them - and extract whatever you can about the general common threads that link the problems. that will take time, but the knowledge thus gained will be pure gold.