Hi All,
Right from the beginning, there are 2 things that you should note about this DS prompt:
1) At NO POINT does it state that K has to be an integer.
2) It's clearly based on exponents, so some exponent rules/patterns MUST be involved.
We're asked if 5^K is < 1,000. This is a YES/NO question.
Fact 1: 5^(K+1) > 3,000
In this Fact, notice how the exponent (K+1) differs from the exponent in the question (K). There's an exponent rule that accounts for this difference.
As an example, consider...
5^2 = 25
5^3 = 125
Notice how 5^3 is "5 times" greater than 5^2? This difference occurs because the base is 5 and we're increasing the exponent by 1. It can also be used in reverse....
5^3/5^2 = 5^(3-2) = 5^1 = 5
This is a standard rule about "dividing" exponents with the same base --> we SUBTRACT the exponents.
With Fact 1, we're dealing with 5^(K+1) and the question is dealing with 5^K. This means that DIVIDING 5^(K+1) by 5 will give us 5^K:
5^(K+1)/5^1 = 5^(K+1-1) = 5^K.
This is all meant to say that we can DIVIDE both sides of this inequality by 5, which gives us...
5^(K+1) > 3,000
5^K > 600
IF....
5^K = 601 then the answer to the question is YES
5^K = 1,001 then the answer to the question is NO
Fact 1 is INSUFFICIENT
Fact 2: 5^(K-1) = 5^K - 500
This is a 1 variable, 1 equation "system", so we CAN solve it (and there will only be 1 answer). Even if you did not know that, it's still easy enough to get to the solution.... Since most Test Takers are better at basic multiplication than they are at manipulating higher-level exponents, here's how you can "brute force" the solution:
5^1 = 5
5^2 = 25
5^3 = 125
5^4 = 625
5^5 = 3125
Find two consecutive powers of 5 that differ by 500 and you have the solution to the above equation.
5^4 - 5^3 = 625 - 125 = 500
Fact 2 is SUFFICIENT.
Final Answer:
GMAT assassins aren't born, they're made,
Rich
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