# GMAT Challenge Series (700+): Q27

Question:

Abe, Beth, Carl and Duncan are four siblings, among which Abe and Carl are twins and Beth and Duncan are also twins. When the present ages of the four siblings are multiplied, the product is 900. If Beth is older than Abe, what is the age of Duncan? Assume the ages of all siblings to be integers.

(1) The difference between Beth’s age and Abe’s age is a prime number.
(2) If Carl had been born four years earlier, the difference between Duncan’s age and Carl’s age would have been a prime number.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) Each statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.

$$\text{Let the age of Abe and Carl be: } \textit{x}$$ $$\text{Let the age of Beth and Duncan be: } \textit{y}$$ $$\text{We know that:}$$ $$\textit{y} > \textit{x}$$ $$\textit{x}^2 \textit{y}^2 = {900}$$ $$\textit{xy} = {30}$$ $$\text{Therefore, possible values for (} \textit{x} \text{, } \textit{y} \text{) are:}$$ $$\text{(1, 30)}$$ $$\text{(2, 15)}$$ $$\text{(3, 10)}$$ $$\text{(5, 6)}$$ $$\text{To answer the question we need to know which combination is correct}$$  $$\text{Statement 1:}$$ $$\text{We are told that } \left| \textit{y} - \textit{x} \right| \text{ is a prime number}$$ $$\text{(1, 30): } \left| \textit{y} - \textit{x} \right| = {29} \text{ (PRIME)}$$ $$\text{(2, 15): } \left| \textit{y} - \textit{x} \right| = {13} \text{ (PRIME)}$$ $$\text{(3, 10): } \left| \textit{y} - \textit{x} \right| = {7} \text{ (PRIME)}$$ $$\text{(5, 6): } \left| \textit{y} - \textit{x} \right| = {1} \text{ (NOT PRIME)}$$  $$\text{There are 3 different values for Duncan's age}$$ $$\text{Statement 1 is NOT SUFFICIENT}$$  $$\text{Statement 2:}$$ $$\text{We are told that } \left| {\textit{y} - \left( {\textit{x} + {4}} \right)} \right| \text{ is a prime number}$$ $$\text{(1, 30): } \left| {\textit{y} - \left( {\textit{x} + {4}} \right)} \right|= {25} \text{ (NOT PRIME)}$$ $$\text{(2, 15): } \left| {\textit{y} - \left( {\textit{x} + {4}} \right)} \right|= {9} \text{ (NOT PRIME)}$$ $$\text{(3, 10): } \left| {\textit{y} - \left( {\textit{x} + {4}} \right)} \right|= {3} \text{ (PRIME)}$$ $$\text{(5, 6): } \left| {\textit{y} - \left( {\textit{x} + {4}} \right)} \right| = {3} \text{ (PRIME)}$$  $$\text{There are 2 different values for Duncan's age}$$ $$\text{Statement 2 is NOT SUFFICIENT}$$  $$\text{Statements 1 & 2 together:}$$ $$\text{(3, 10) is the only combination which holds true for statements 1 and 2}$$ $$\text{Statements 1 & 2 together are SUFFICIENT}$$