Is the integer b divisible by 3 ?

Statement 1: Considering 8 is divisible by 2, in order for 8b to be divisible by 6, b needs to be divisible by 3. Hence Sufficient.
Statement 2: Since 9b is already divisible by 3, b may or may not be divisible by 3. Not sufficient.

Statement 1: 8b : If it is to divisible by 6 , it has to have factors (3 and 2). 8 can provide factor of 2, not 3. So, b has to have factor of 3. SUFFICIENT.

Statement 2: 9*b : 9 has 3*3; b need not have factor of 3 for 9*b to be divisible by 12.

not SUFFICIENT.

Ans. - A

\(b\,\,\operatorname{int}\)

\(\frac{b}{3}\,\,\mathop = \limits^? \,\,\operatorname{int}\)

\(\left( 1 \right)\,\,\frac{{4b}}{3}\,\, = \,\,\,\operatorname{int} \,\,\,\,\mathop \Rightarrow \limits^{GCD\left( {3,4} \right)\,\, = \,\,1} \,\,\,\frac{b}{3} = \operatorname{int} \,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle\)

\(\left( 2 \right)\,\,\frac{{3b}}{4}\,\, = \,\,\,\operatorname{int} \,\,\,\,\left\{ \begin{gathered}\\
\,{\text{Take}}\,\,b = 0\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \hfill \\\\
\,{\text{Take}}\,\,b = 4\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\ \\
\end{gathered} \right.\)

This solution follows the notations and rationale taught in the GMATH method.

Statement 1:
4b is divisible by 3. The factor of 3 can come from either 4 or b, since 3 is not a factor of 4 it must be a factor of b. Therefore b is divisible by 3. Sufficient.

Statement 2:
3b is divisible by 4. Again from the logic above, b has a factor of 4 but we still don't know if it has a factor of 3. Insufficient.

Ans: A

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