When the positive integer n is divided by 25, the remainder

[swerve]

When the positive integer n is divided by 25, the remainder is 13. What is the value of n?

1. n < 100
2. When n is divided by 20, the remainder is 3.

The OA is C.

Source: GMAT Prep

[Brent@GMATPrepNow]

When the positive integer n is divided by 25, the remainder is 13. What is the value of n?

1. n < 100
2. When n is divided by 20, the remainder is 3.

Target question: What is the value of n?

Given: When the positive integer n is divided by 25, the remainder is 13.
-------ASIDE-------------
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.
------BACK TO THE QUESTION----------------
So, from the given information, the possible values of n are: 13, 38, 63, 88, 113, . . .

Statement 1: n < 100
We already know that the possible values of n are: 13, 38, 63, 88, 113, . . .
So, if n < 100, then n can equal 13, 38, 63, or 88
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: When n is divided by 20, the remainder is 3.
We already know that the possible values of n are: 13, 38, 63, 88, 113, . . .
We can see that 63 meets the condition that, when n is divided by 20, the remainder is 3.
Is that the only value?
Let's list a few more possible n-values to get: ...113, 138, 163
STOP
163 also meets the condition that, when n is divided by 20, the remainder is 3.
At this point, we can see that n can equal 63 or 163 (and quite possibly some other values)
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 indirectly tells us that the possible values of n are 13, 38, 63 or 88
Among those possible n-values, only 1 value (63) satisfies statement 2
So, the answer to the target question is n = 63
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent

[fskilnik@GMATH]

When the positive integer n is divided by 25, the remainder is 13. What is the value of n?

1. n < 100
2. When n is divided by 20, the remainder is 3.
Source: GMAT Prep

$$\left\{ \matrix{
n \ge 1\,\,{\mathop{\rm int}} \hfill \cr
n = 25Q + 13,\,\,Q \ge 0\,\,{\mathop{\rm int}} \hfill \cr} \right.$$
$$? = n$$
$$\left( 1 \right)\,\,n < 100\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,{\rm{Q = 0}}\,\,\,\, \Rightarrow \,\,\,\,\,n = 13 \hfill \cr
\,{\rm{Take}}\,\,{\rm{Q = 1}}\,\,\,\, \Rightarrow \,\,\,\,\,n = 38 \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\left\{ \matrix{
n = 20K + 3,\,\,K \ge 0\,\,{\mathop{\rm int}} \hfill \cr
n = 25Q + 13,\,\,Q \ge 0\,\,{\mathop{\rm int}} \hfill \cr} \right.\,\,\,\, \Rightarrow \,\,\,25Q + 10 = n - 3\,\, = \,\,20K$$
$$ \Rightarrow \,\,\,25Q + 10\,\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,20\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,Q = 2\,\,\, \Rightarrow \,\,? = n = 63 \hfill \cr
\,{\rm{Take}}\,\,Q = 6\,\,\, \Rightarrow \,\,? = n = 163 \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\,? = n = 63$$

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

Regards,
Fabio.