Problem Solving

Manhattan Prep

When the positive integer x is divided by 9, the remainder is 5. What is the remainder when 3x is divided by 9?

A. 0
B. 1
C. 3
D. 4
E. 6

OA E.

AAPL wrote:Manhattan Prep

When the positive integer x is divided by 9, the remainder is 5. What is the remainder when 3x is divided by 9?

A. 0
B. 1
C. 3
D. 4
E. 6

Let x=5, since dividing 5 by 9 yields a remainder of 5:
5/9 = 0 R5.
When 3x=15 is divided by 9, we get:
15/9 = 1 R6.

The correct answer is E.

AAPL wrote:Manhattan Prep

When the positive integer x is divided by 9, the remainder is 5. What is the remainder when 3x is divided by 9?

A. 0
B. 1
C. 3
D. 4
E. 6

$$x \ge 1\,\,\,{\mathop{\rm int}} \,\,\,\left( * \right)$$
$$\left\{ \matrix{
3x = 9J + R\,, \hfill \cr
0 \le R\,\,{\mathop{\rm int}} \,\le 8\,\,\,,\,\,J\,\,\mathop \ge \limits^{\left( * \right)} \,\,0\,\,\,{\mathop{\rm int}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\,;\,\,\,\,\,\,\,\,? = R$$
$$x = 9Q + 5\,\,,\,\,\,Q\,\,\mathop \ge \limits^{\left( * \right)} \,\,0\,\,\,{\mathop{\rm int}} \,\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,\,3} \,\,\,\,\,3x = 9\left( {3Q} \right) + 9 + 6 = 9\left( {3Q + 1} \right) + 6\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,? = 6\,\,\,\,\,\,\,\left[ {\,{\rm{and}}\,\,J = 3Q + 1\,} \right]$$


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

Regards,
Fabio.