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.