# GMAT Challenge Series (700+): Q23

Question:

If a, b, and c are integers such that 0 < a < b < c < 10, is the product abc divisible by 3?

(1) If a/1000 + b/100 + c/10 is expressed as a single fraction reduced to lowest terms, the denominator is 200.
(2) c – b < b – a

(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.

$$\textit{abc} \text{ will be divisible by 3 if } \textit{a} \text{, } \textit{b} \text{, or} \textit{c} \text{ equals 3, 6, or 9}$$  $$\text{Statement 1:}$$ $${\textit{a} \over {1000}} + {\textit{b} \over {100}} + {\textit{c} \over {10}} = {{\textit{a} + {10}\textit{b} + {100}\textit{c}} \over {1000}} = {{{\textit{a} + {10}\textit{b} + {100}\textit{c}} \over {5}} \over {200}}$$ $$\textit{a} + {10}\textit{b} + {100}\textit{c} \text{ must be divisible by 5}$$ $$\text{This implies that } \textit{a} \text{ must be divisible by 5}$$ $$\text{Since } {0} < \textit{a} < {10}, \textit{a} = {5}$$ $$\textit{abc} \text{ wont be divisible by 3 if, and only if, } \textit{b} \text{ and } \textit{c} \text{ are 7 and 8 respectively}$$ $$\text{In this case:}$$ $${{\textit{a} + {10}\textit{b} + {100}\textit{c}} \over {1000}} = \frac{875}{1000} = \frac{7}{8}$$ $$\text{Reduced to the lowest terms the denominator is 8 and not 200}$$ $$\text{Therfore, } \textit{b} \text{ must be 6 or } \textit{c} \text{ must be 9}$$ $$\text{Statement 1 is SUFFICIENT}$$  $$\text{Statement 2:}$$ $$\textit{c} - \textit{b} < \textit{b} - \textit{a}$$ $$\textit{a} + \textit{c} < {2}\textit{b}$$ $$\text{Expression valid for: } \textit{a} = {1} \textit{ b} = {5} \textit{ c} = {7} \text{ where} \textit{ abc} \text{ is not divisible by 3}$$ $$\text{Expression valid for: } \textit{a} = {1} \textit{ b} = {6} \textit{ c} = {7} \text{ where} \textit{ abc} \text{ is divisible by 3}$$ $$\text{Statement 2 is NOT SUFFICIENT}$$