Arithmetic Properties of numbers Quant Review 2nd Ed #68

[runningguy]

When positive integer n is divided by 5, the remainder is 1. When n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 35?

A) 3
B) 4
C) 12
D) 32
E) 35

I got the answer by plugging in numbers for n until I found the number (31) that met the remainder requirements, but I feel like I won't have the time and it is a risky method for the test. What is the easiest way to tackle a problem like this in under 2 minutes without guessing numbers? The explanation in the guide is difficult for me to understand.

One more question, how many questions can we post up at one time? I have a lot more questions but I don't want to flood the forum. Thanks!

[Brent@GMATPrepNow]

When positive integer n is divided by 5, the remainder is 1. When n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 35?

A) 3
B) 4
C) 12
D) 32
E) 35

I got the answer by plugging in numbers for n until I found the number (31) that met the remainder requirements, but I feel like I won't have the time and it is a risky method for the test. What is the easiest way to tackle a problem like this in under 2 minutes without guessing numbers? The explanation in the guide is difficult for me to understand.

One more question, how many questions can we post up at one time? I have a lot more questions but I don't want to flood the forum. Thanks!

I'm not sure what you mean by "plugging in numbers," but there is a systematic approach to listing possible values.

There's a nice rule that says, If, when N is divided by D, the remainder is R, then the possible values of N include: R, R+D, R+2D, R+3D,. . .

When n is divided by 5, the remainder is 1.
So, possible values of n are 1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, 61, 66, 71, 76, etc.

When n is divided by 7, the remainder is 3.
So, possible values of n are 3, 10, 17, 24, 31, 38, 45, 52, 59, 66, 73, etc.

So, we can see that n could equal 31, or 66, or an infinite number of other values.

Important: Since the Least Common Multiple of 7 and 5 is 35, we can conclude that if we list the possible values of n, each value will be 35 greater than the last value.

So, n could equal 31, 66, 101, 136, and so on.

Answer choice A: If we add 3 to any of these possible n-values, the sum is NOT a multiple of 35.
ELIMINATE A

Answer choice B: if we take any of these possible n-values, and add 4, the sum will be a multiple of 35.

So, the smallest value of k is 4 such that k+n is a multiple of 35.

Answer = B

Cheers,
Brent

[runningguy]

Brent,

Thank you for the explanation. I had not heard of that rule before but it will get put to good use. This makes much more sense than the explanation given in the answer key.

What I meant by plugging in numbers is that I knew n/5 had to have R1 and n/7 had to have R3. Therefore I started with 6, which gave me the correct remainder for n/5 but it did not for n/7. Then I plugged in 11 for n, which again gave me the correct remainder for n/5 but not for n/7. I did this until I got to 31 which met both criteria and gave the correct remainders for each scenario. Then I added 4 to make it a factor of 35.

[Brent@GMATPrepNow]

Your method looks great.
Once you become fast at listing possible values, you can solve a question like this in under 1 minute.

Cheers,
Brent

[[email protected]]

Hi runningguy,

The approach that Brent showed you is a great way to solve this type of question; it's exactly the way that I'd approach it. For all intents and purposes, it's the best/fastest way to approach this question.

You mentioned one other thing in your original post, which I think needs to be addressed. While the AVERAGE amount of time per Quant question is about 2 minutes (75 minutes/37 questions = about 2 minutes per question), you should NOT expect to spend 2 minutes on each question. Certain questions can be answered relatively quickly (in under a minute), others require 3 minutes (even if you know the shortcuts). So make sure to be flexible with how much time you spend on a given question.

GMAT assassins aren't born, they're made,
Rich

[ganeshrkamath]

When positive integer n is divided by 5, the remainder is 1. When n is divided by 7, the remainder is 3. What is the smallest positive integer k such that k+n is a multiple of 35?

A) 3
B) 4
C) 12
D) 32
E) 35

I got the answer by plugging in numbers for n until I found the number (31) that met the remainder requirements, but I feel like I won't have the time and it is a risky method for the test. What is the easiest way to tackle a problem like this in under 2 minutes without guessing numbers? The explanation in the guide is difficult for me to understand.

One more question, how many questions can we post up at one time? I have a lot more questions but I don't want to flood the forum. Thanks!

n = 5a + 1 = 7b + 3
5a = 7b + 2
a = (7b + 2)/5
Find values of b that result in integer values for a.
(a,b) = (6,4), (13,9), (20,14), ....
n = 31, 66, 101, ...
To make (n+k) a multiple of 35:
k = 04, 04, 04, ...

Choose B

Cheers