Data Sufficiency Pack 4, Question 2) Z is a positive integer...

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.

Z is a positive integer greater than 3. How many distinct prime factors does (Z + 1)(Z – 1) have?

1) Z is not even
2) Z is not a multiple of 5

There is one variable (z) and 2 equations are given by the 2 conditions, increasing the chance (D) will be our answer.
For condition 1, z=3, (3+1)(3-1)=8=2^3, the no. of distinct prime factors:1
z=7, (7+1)(7-1)=(2^4)3, the no. of distinct prime factors: 2
For condition 2, z=3, (3+1)(3-1)=8=2^3, the no. of distinct prime factors:1
z=7, (7+1)(7-1)=(2^4)3, the no. of distinct prime factors: 2
Looking at the conditions together,
z=3, (3+1)(3-1)=8=2^3, the no. of distinct prime factors:1
z=7, (7+1)(7-1)=(2^4)3, the no. of distinct prime factors: 2.
Therefore, the answer becomes (E).

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.