When a positive integer n is divided by 19, what is the rema

[Max@Math Revolution]

[Math Revolution GMAT math practice question]

When a positive integer n is divided by 19, what is the remainder?

1) n-17 is a multiple of 19
2) n-19 is a multiple of 17

[fskilnik@GMATH]

[Math Revolution GMAT math practice question]

When a positive integer n is divided by 19, what is the remainder?

1) n-17 is a multiple of 19
2) n-19 is a multiple of 17

$$1 \le n = 19Q + R\,\,,\,\,0 \le R \le 18\,\,\,\,\,\left( {Q,R\,\,{\rm{ints}}} \right)$$
$$? = R$$

$$\left( 1 \right)\,\,n - 17 = 19M\,\,,\,\,M \ge 0\,\,{\mathop{\rm int}} $$
$$n = 19M + 17\,\,\, \Rightarrow \,\,\,\left\{ \matrix{
\,Q = M \hfill \cr
\,? = R = 17\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}. \hfill \cr} \right.$$

$$\left( 2 \right)\,\,n - 19 = 17K\,\,,\,\,K \ge - 1\,\,{\mathop{\rm int}} \,\,\, \Rightarrow \,\,\,n = 17\left( {K + 1} \right) + 2$$
$$ \Rightarrow \,\,\,n = 17J + 2\,\,,\,\,J\left( { = K + 1} \right) \ge 0\,\,{\mathop{\rm int}} \,$$
$$\left\{ \matrix{
\,{\rm{Take}}\,\,{\rm{J = 0}}\,\,\, \Rightarrow \,\,\,n = 2\,\,\, \Rightarrow \,\,\,? = R = 2\,\,\,\,\,\left( {Q = 0} \right) \hfill \cr
\,{\rm{Take}}\,\,{\rm{J = 1}}\,\,\, \Rightarrow \,\,\,n = 19\,\,\, \Rightarrow \,\,\,? = R = 0\,\,\,\,\,\left( {Q = 1} \right) \hfill \cr} \right.$$

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

Regards,
Fabio.

[Brent@GMATPrepNow]

[Math Revolution GMAT math practice question]

When a positive integer n is divided by 19, what is the remainder?

1) n-17 is a multiple of 19
2) n-19 is a multiple of 17

Target question: When n is divided by 19, what is the remainder?

Statement 1: n-17 is a multiple of 19
----ASIDE-------------------------------------------
If N is a multiple of d, then we can write N = dk (for some integer k)
For example, if N is a multiple of 5, then we can write N = 5k (for some integer k)
----BACK TO THE QUESTION-------------------

If n-17 is a multiple of 19, then we can write: n - 17 = 19k
Add 17 to both sides to get: n = 19k + 17
In other words, n is 17 greater than some multiple of 19
So, when we divide n by 19, the remainder will be 17
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: n-19 is a multiple of 17
We can write n-19 = 17k
Add 19 to both sides to get: n = 17k + 19
Hmmm, this information doesn't help us answer the target question
Consider these two contradictory cases:
Case a: if k = 1, then n = 17(1) + 19 = 36. When we divide 36 by 19, the answer to the target question is the remainder is 17
Case b: if k = 2, then n = 17(2) + 19 = 53. When we divide 53 by 19, the answer to the target question is the remainder is 15
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent

[Max@Math Revolution]

=>

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 1 variable (n) and 0 equations, D is most likely to be the answer. So, we should consider each of the conditions on its own first.

Condition 1)
n - 17 = 19*k for some integer k.
So, n = 19*k + 17, which means that n has a remainder of 17 when it is divided by 19.
Condition 1) is sufficient.

Condition 2)
Now, n - 19 = 17*m or n = 17*m + 19.
When m = 1, n = 36, and so n = 19*1 + 17 has a remainder of 17 when it is divided by 19.
When m = 2, n = 53, and so n = 19*2 + 15 has a remainder of 15 when it is divided by 19.
Since we don't have a unique remainder, condition 2) is not sufficient.

Therefore, A is the answer.
Answer: A

If the original condition includes "1 variable", or "2 variables and 1 equation", or "3 variables and 2 equations" etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.