When positive integer n is divided by 13, the remainder is 2

[BTGmoderatorLU]

Veritas Prep

When positive integer n is divided by 13, the remainder is 2. When n is divided by 8, the remainder is 5. How many such values are less than 180?

A. 0
B. 1
C. 2
D. 3
E. 4

OA B

[fskilnik@GMATH]

Veritas Prep

When positive integer n is divided by 13, the remainder is 2. When n is divided by 8, the remainder is 5. How many such values are less than 180?

A. 0
B. 1
C. 2
D. 3
E. 4

$$1 \le n \le 179\,\,\,\,\,\left( {n\,\,\,{\mathop{\rm int}} } \right)\,\,\,\,\left( * \right)$$
$$n = 13M + 2\,\,\,\,,\,\,\,M\,\,{\mathop{\rm int}} \,\,\,\,\left( {\rm{I}} \right)$$
$$n = 8J + 5\,\,\,\,,\,\,\,J\,\,{\mathop{\rm int}} \,\,\,\,\left( {{\rm{II}}} \right)$$
\[?\,\,\,:\,\,\,n\,\,\,{\text{in}}\,\,\,\left( * \right) \cap \left( {\text{I}} \right) \cap \left( {{\text{II}}} \right)\]

$$\left( * \right)\,\, \cap \,\,\left( {\rm{I}} \right)\,\,\,:\,\,\,\,\,\,1\,\, \le \,\,13M + 2\,\, \le \,\,179\,\,\,\,\,\mathop \Leftrightarrow \limits^{ - \,2} \,\,\,\,\, - 1 \le 13M \le 177\,\,\left( { = 169 + 8} \right)$$
$$ - 1 \le 13M \le 177\,\,\left( { = 169 + 8} \right)\,\,\,\,\,\mathop \Leftrightarrow \limits^{M\,\,{\mathop{\rm int}} } \,\,\,0 \le 13M \le 169\,\,\,\,\,\mathop \Leftrightarrow \limits^{:\,\,13} \,\,\,\,0 \le M \le 13$$

$$\left( * \right)\,\, \cap \,\,\left( {{\rm{II}}} \right)\,\,\,:\,\,\,\,\,\,\left\{ \matrix{
\,n - 5\,\,{\rm{divisible}}\,\,{\rm{by}}\,\,8\,\,\,\,\, \Rightarrow \,\,\,\,\,n - 5\,\,\,\,\,{\rm{even}}\,\,\,\,\, \Rightarrow \,\,\,\,\,n\,\,\,{\rm{odd}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left( {\rm{I}} \right)} \,\,\,\,\,M\,\,{\rm{odd}} \hfill \cr
\,n - 5\mathop = \limits^{\left( {\rm{I}} \right)} 13M - 3\,\,{\rm{divisible}}\,\,{\rm{by}}\,\,8\,\,\,\,\, \Rightarrow \,\,\,\,\,{{13M - 3} \over 2}\,\,\,\,{\rm{divisible}}\,\,{\rm{by}}\,\,4\,\,\,\left( {***} \right)\,\,\,\,\,\, \hfill \cr} \right.$$
\[\left. \begin{gathered}
M = 1\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 - 3}}{2} = 5\,\,\,{\text{odd}}\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
M = 3\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 \cdot 3 - 3}}{2} = 18\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
M = 5\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 \cdot 5 - 3}}{2} = 31\,\,{\text{odd}}\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}}\,\,\,\,\,\, \hfill \\
\boxed{M = 7}\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 \cdot 7 - 3}}{2} = 44\,\,\,\,\left( {***} \right)\,\,\,{\text{YES}} \hfill \\
M = 9\,\,\,\,\, \Rightarrow \,\,\,\,{\text{odd}}\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
M = 11\,\,\,\,\, \Rightarrow \,\,\,\,\frac{{13 \cdot 11 - 3}}{2} = 70\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
M = 13\,\,\,\,\, \Rightarrow \,\,\,\,{\text{odd}}\,\,\,\,\left( {***} \right)\,\,\,{\text{NO}} \hfill \\
\end{gathered} \right\}\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 1\]

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

Regards,
Fabio.

[Brent@GMATPrepNow]

Veritas Prep

When positive integer n is divided by 13, the remainder is 2. When n is divided by 8, the remainder is 5. How many such values are less than 180?

A. 0
B. 1
C. 2
D. 3
E. 4

When it comes to remainders, we have a nice rule that says:
If N divided by D leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

When positive integer n is divided by 13, the remainder is 2.
So, some possible values of n (less than 180) are: 2, 15, 28, 41, 54, 67, 80, 93, 106, 119, 132, 145, 158, 171

When n is divided by 8, the remainder is 5.
Now check the above list of numbers and identify all of the values that, when divided by 8, leave a remainder of 5
I've highlighted in red all that apply

Answer: B

Cheers,
Brent

[Scott@TargetTestPrep]

Veritas Prep

When positive integer n is divided by 13, the remainder is 2. When n is divided by 8, the remainder is 5. How many such values are less than 180?

A. 0
B. 1
C. 2
D. 3
E. 4

We can create the equation:

n = 13Q + 2

So n can be 2, 15, 28, 41, 54, 67, 80, 93, ...

and

n = 8R + 5

So n can be 5, 13, 21, 29, 37, 45, 53, 61, 69, 77, 85, 93, ...

We see that the first number that satisfies both conditions is 93. To find the other numbers, we can keep adding the LCM of 13 and 8, which is 13 x 8 = 104. Therefore, the next value that satisfies both conditions is 93 + 104 = 197. However, 197 is already greater than 180, so we only have one value, namely 93, that is less than 180 and satisfies both conditions.

Alternate Solution:

Since the remainder when n is divided by 13 is 2, n = 13Q + 2 for some Q. Similarly, since the remainder when n is divided by 8 is 5, n = 8R + 5 for some R. Then, n + 11 = 13Q + 13 = 8R + 16 is divisible by both 8 and 13. Thus, n + 11 is divisible by the LCM of 8 and 13, which is 104. If n + 11 = 104, then n = 93. If n + 11 = 208, then n = 197, which is already greater than 180. Thus, there is only one possible value of n under 180.

Answer: B