110 Quant2nd Edition

If

n is an integer between 2 and 100 and if

n is also the square of an integer, what is the value of

n?

(1) n is even

(2) the cube root of n is an integerMy thoughts

n is an integer between 2 and 100... (keep reading von, don't make a careless error because you prematurely solved the problem without reading the question) ... and

n is the square of an integer so let's make a chart and test values:

integer.....n.........2<n<100?

1.............1.........NO

2.............4.........YES

3.............9.........YES

4.............16.......YES

5.............25.......YES

6.............36.......YES

7.............49.......YES

8.............64.......YES

9.............81.......YES

10...........100.......NO

So

n could be 4, 9, 16, 25, 36, 49, 64, or 81 since each of these values represent the square of an integer between 2 and 100 exclusive.

(1)n is evenWell from my above chart, we can see at least two unique values for n when n is even, (For example: 16 and 36); therefore, this statement is insufficient to answer the question because it can have more than one value. In my answer grid I eliminated

[strike]AD[/strike]BCE(2)the cube root of n is an integerMake a chart again:

n..............\(\sqrt[3]{n}\)

8................2

27..............3

So \(\sqrt[3]{n}\) could give you different values depending on the value of n; therefore, this statement is insuffcient as well because it yields more than one answer for n.

[strike]ADB[/strike]CECombining statements (1) and (2) together I got the following table:

n..............\(\sqrt[3]{n}\)

8................2

27...............3

64...............4

So we see that

n is even per statement (1) and the cube root of

n is an integer for 8, 27, and 64; thus there three possible values for

n; thus statements (1) and (2) together are not sufficent to answer the question. [strike]

ADBC[/strike]E; therefore, the answer must be

.

When considering the second statement you forgot that n must also be a perfects square.

Given: n is a perfect square between 2 and 100 (a perfect square is an integer that can be written as the square of some other integer, for example 16=4^2, is a perfect square).

(1) n is even --> n can be any even perfect square in the given range: 4, 16, 36, ... Not sufficient.

(2) The cube root of n is an integer --> so n is also a perfect cube between 2 and 100. There are 4 perfect cubes in this range: 2^3=8, 3^3=27 and 4^3=64 but only one of them namely 64 is also a perfect square, so n=64=8^2=4^3. Sufficient.