Max@Math Revolution wrote:N is a positive integer. Is n the square of an integer?

1) 4n is the square of an integer
2) n^3 is the square of an integer


* A solution will be posted in two days.

Question : Is n the square of an integer?

Statement 1: 4n is the square of an integer
Since 4 is already a perfect square of an Integer so n must be a perfect square (Given:n is Integer)
e.g. for 4n = 4, n = 1 which is square of Integer and n also is square of an Integer (YES)
e.g. for 4n = 16, n = 4 which is square of Integer and n also is square of an Integer (YES)
SUFFICIENT

Statement 2: n^3 is the square of an integer
for n^3 = 1, n = 1 which is square of Integer(YES)
for n^3 = 4, n = Not an Integer so not acceptable
for n^3 = 9, n = Not an Integer so not acceptable
for n^3 = 16, n = Not an Integer so not acceptable
for n^3 = 25, n = Not an Integer so not acceptable
for n^3 = 36, n = Not an Integer so not acceptable
for n^3 = 49, n = Not an Integer so not acceptable
for n^3 = 64, n = 4 i.e. a perfect square (YES)
NOTE: for n to be integer and n^3 to be a square the number should be of the form [m]a^6[/m]
i.e. n must be a perfect square
SUFFICIENT


Answer: option D

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

N is a positive integer. Is n the square of an integer?
1) 4n is the square of an integer
2) n^3 is the square of an integer


In the original condition, there is 1 variable(n), which should match with the number of equation. So you need 1 equation. For 1) 1 equation, for 2) 1 equation, which is likely to make D the answer.
For 1), in 4n=m^2 (m is some integer), 4n is an even number so it should be m^2=even. Then, m^2=(2k) ^2 where k is some integer and therefore 4n=(2k)^2=4k^2 -> n=k^2, which is yes and sufficient.
For 2), n=3√(t^2 ) is derived from n^3=t^2 where t is some integer. Since n is a positive integer, cube root should be removed. Therefore, n=3√(t^2 )=3√{(s^3)}^2 =3√(s^6 )=s^2 is also yes and sufficient. Therefore, the answer is D.


� For cases where we need 1 more equation, such as original conditions with "1 variable", or "2 variables and 1 equation", or "3 variables and 2 equations", we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.