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What is the value of the integer n

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Source: — Data Sufficiency |
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by Vincen » Sun Mar 11, 2018 7:28 am
Hello.

I would solve this as follows:

(1) n(n+2)=15.

This is not sufficient because this contidition implies that n=3 or n=-5, because $$3\left(3+2\right)=3\cdot5=15$$ $$-5\left(-5+2\right)=-5\cdot\left(-3\right)=15$$ Hence, this statement is not sufficient to determine the value of n. INSUFFICIENT.

(2) (n+2)^n = 125.

Now if we rewrite this expression we get that $$125=125^1\ \ \ \ \ \ or\ \ \ \ \ \ \ \ 125=5^3=\left(3+2\right)^3.$$ Hence, we get from this that n=3. Therefore, this statement is SUFFICIENT.

Thus the answer is the option B.
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by ErikaPrepScholar » Wed Mar 14, 2018 10:06 am
We can also think about the statements slightly differently:

Statement 1
$$n\left(n+2\right)=15$$ $$n^2+2n=15$$ $$n^2+2n-15=0$$ $$\left(n+5\right)\left(n-3\right)=0$$ $$n=-5\ or\ n=3$$
So there are two possible values of n. Insufficient.

Statement 2
If n is an integer, then n+3 is an integer. If we raise an integer (positive or negative) to a negative power, the result will be a fraction between 0 and 1 (if the integer is positive) or between -1 and 0 (if the integer is negative). We know that raising n+3 to the power of n is a whole number (125), so we know that n cannot be negative.
We also know that n cannot be 0, since anything raised to the power of zero is 1. So n must be a positive integer.
We can test positive values for n until we reach our answer. Since n is our exponent, we know that the value of the expression will increase rapidly with n, so we shouldn't need to test many numbers - the expression will get much too large quickly:
$$n=1\ \ -->\left(1+2\right)^1=3^1=3$$ $$n=2\ -->\left(2+2\right)^2=4^2=16$$ $$n=3\ -->\left(3+2\right)^3=5^3=125$$
So n must be 3. Sufficient.
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