# Lesson: Choosing Good Numbers

## Comment on Choosing Good Numbers

### by choosing 1 you say that 1

by choosing 1 you say that 1/5 gives remainder 1...i am not able to understand it as 1/5 is 0.2 and no remainder.please explain me.

### I can see how you might think

I can see how you might think that. However, if we think about remainders in the way you suggest, there would never been any remainders. For example, we'd just say that 7/2 = 3.5 (no remainder), when in fact, 2 divides into 7 three times with remainder 1.

Likewise, 1 divides into 5 zero times, with remainder 1.

For more on this, watch the video on remainders: https://www.gmatprepnow.com/module/gmat-integer-properties/video/842

### hi.. this is sujan and i have

hi.. this is sujan and i have a doubt of one example in good numbers video. it is a data sufficiency example. that question is k+5/k is an integer?? in that question 1/5 leaves remainder 1 u said.. how is it possible?? can u explain again that question??

### Good question.

Good question.

Some background information//examples first:
a) 3 divides evenly into 6 two times
b) 5 divides evenly into 20 four times
c) 2 divides evenly into 14 seven times

If a number does NOT divide evenly into another number then we have a REMAINDER.

Some examples:
d) 3 does not divide evenly into 17, so we have a remainder. 3 divides into 17 five times with remainder 2.
So, we can write: 17 = (5)(3) + 2

e) 2 does not divide evenly into 23, so we have a remainder. 2 divides into 23 eleven times with remainder 1.
So, we can write: 23 = (11)(2) + 1

5 does not divide evenly into 1, so there must be a remainder. What is that remainder?

5 divides into 1 ZERO times with remainder 1.
So, we can write: 1 = (0)(5) + 1

### In the video, "When K is

In the video, "When K is divided by 5", shouldn't be 4? May be the context is “When 5 is divided by K”?

### Hi linnn01,

Hi linnn01,

I'm not sure what you are asking.
Are you suggesting that (for statement 1) k could equal 4?
If so, then this is not correct, because 4 divided by 5 is 0 with remainder 4.

Please let me know if this is what you meant.

### Ref. to the second to last

Ref. to the second to last question in the video, we can rephrase the target question to ask : 'is k a multiple of 5?'. Similarly for the last question, we can rephrase the target question to ask:'Does k equal 2?' Please let me know if my approach is correct. Thanks

### Your post: We can rephrase

Your post: We can rephrase the target question to ask : 'Is k a multiple of 5?'

The target question: Is (k + 5)/k an integer?
Notice that (k + 5)/k = k/k + 5/k = 1 + 5/k
So, the expression will be an integer if k = 5 OR if k = 1

So, the rephrased target question could be "Is k EITHER equal to 5 OR equal to 1?

----------------------------------
Your post: Similarly for the last question, we can rephrase the target question to ask:'Does k equal 2?

No, this would not be a correct rephrasing of the target question.

The target question: Is (k+1)/2 an integer?

In order for (k+1)/2 to be an integer, it must be the case that (k+1) is EVEN.

If (k+1) is EVEN, then it must be the case that k is ODD.

So, we can rephrase the target question to ask "Is k odd?"

Cheers,
Brent

### Hi Brent,

Hi Brent,

Please refer to this OG question : https://gmatclub.com/forum/if-x-y-and-z-are-positive-numbers-what-is-the-value-of-the-average-ar-243301.html

I was able to combine both statements and arrive at (y+1)^2 = x^2 + x + 1
At this step I got confused and started wondering if there might be a unique value of x that might be picked to get a unique value for y. This wasted precious time. Please suggest how to avoid getting confused in such a scenario and how could I have picked numbers/approached the problem in a better way?

Thanks & Regards,
Abhirup

In that situation, you should give yourself 30 seconds (tops!) to see if you can solve the equation (y+1)² = x² + x + 1 for x and y.
If you can't solve the equation, then you need to abandon that strategy and either come up with a new strategy or make an educated guess and move on. That's the key to the GMAT in general.

I hope that helps.

Cheers,
Brent

### Hi Brent, you are the man of

Hi Brent, you are the man of the moment!! Data sufficiency is finally improving for me! :)

Thanks for that!

### That´s a very good video, but

That´s a very good video, but I have one small question. At 4:39 and 6:56, the number we chose got a result, and it has already caused conflicting answers to the target question, why should we still keep choosing more numbers, why can´t we just say it´s insufficient directly please? Thank you and looking for ward to your reply. Have a lovely day!

### In both of the cases you

In both of the cases you mentioned, the target questions are YES/NO questions.

At 4:39 and 6:56, we have only ONE answer to the target question, so we don't yet have CONFLICTING answers to those target questions.

For example, in the first question, the target question asks: Is (x+1)/(x-1) < 0?
Statement 1: x < 0

When we plug in x = -1, we see that the answer to the target question is "NO, (x+1)/(x-1) is NOT less than 0"

At this point, we don't know whether statement 1 is sufficient.
It COULD be the case that (x+1)/(x-1) is NEVER less than 0, in which case statement 1 is sufficient.
Or there COULD instances in which (x+1)/(x-1) is less than 0, AND instances in which (x+1)/(x-1) is NOT less than 0, in which case statement 1 is NOT sufficient.

So, we can't conclude that statement 1 is not sufficient, until we find a value for x that yields a DIFFERENT answer to the target question.

Later, when we plug in x = -0.5, we see that the answer to the target question this time is "YES, (x+1)/(x-1) IS less than 0"

At this point, we have two different answers to the target question (one YES and one NO). So, we can say statement 1 is NOT sufficient.

Does that help?

Cheers,
Brent

### Hi Brent,

Hi Brent,

I have a clarification question around the comment:
So, the rephrased target question could be "Is k EITHER a multiple of 5 OR equal to 1?

Evaluating the expression:
(k+5)/k = k/k + 5/k
= 1+ 5/k

I can think of only two values: 1 and 5 that can make this expression evaluate to an integer. So won't the rephrasing be "is the value of k 1 or 5"? I'm curious to know if there are any other values that'd satisfy this expression.
Thanks!

### Good catch!

Good catch!
You're absolutely right. I have edited my response above.

Cheers,
Brent

### I'm confused: 1 divided by 5

I'm confused: 1 divided by 5 is 0.2. So, the remainder is 2, and not 1 as mentioned in this video? Thanks!

### The process of calculating

The process of calculating remainders is different from what you are suggesting. Later, in the Integer Properties module, you will learn how to calculate remainders: https://www.gmatprepnow.com/module/gmat-integer-properties/video/842

In the meantime, here are a few examples:

7 ÷ 2 = 3 with remainder 1
In other words, 2 divides into 7 three times, leaving a remainder of 1

23 ÷ 4 = 5 with remainder 3
In other words, 4 divides into 23 five times, leaving a remainder of 3

1 ÷ 5 = 0 with remainder 1
In other words, 5 divides into 1 zero times, leaving a remainder of 1

As I mention above (in the Reinforcement Activities box), if you’re viewing this Data Sufficiency lesson early in your preparation, you may be unfamiliar with some of the topics/concepts required to answer the practice questions below. Don’t worry; those topics/concepts are covered in subsequent lessons.

Dear Brent. Please clarify something for me.
The question: Is (x+1)/(x-1) < 0? Statement 1 says x < 0.

Rephrasing the question I have the following:
(x+1)/(x-1) < 0, divide by x-1
x+1 < 0
x < -1
Is x < -1?
Is the question rephrase correct?

Lets say yes then statement 1 is sufficient? What am I missing here?
PS Thank you for your great videos. I like your course more than e-GMAT course which I used before.

### Typically, we rephrase the

Typically, we rephrase the target question BEFORE examining each statement.
That is, we use whatever available information we have to rephrase the target question.

That said, your approach is perfect except for one thing: when we divide both sides of an inequality by a NEGATIVE value, we must REVERSE the direction of the inequality symbol (this is covered later on in the following video: https://www.gmatprepnow.com/node/979/edit?destination=module/gmat-algebr...)

So, if x < 0, we know that x-1 will be negative.
So, when we take (x+1)/(x-1) < 0, and divide both sides by x-1, we get: x + 1 > 0
Now subtract 1 from both sides to get: x > -1

So, we can ask "Is x > -1?"
Since statement 1 tells us that x < 0, it COULD be the case x = -0.5, In which case the answer to your rephrased target question is "YES, x is greater than -1"
Conversely, it COULD also be the case x = -2, In which case the answer to your rephrased target question is "NO, x is not greater than -1"
So statement 1 is not sufficient

Cheers,
Brent
PS: I'm happy to hear you like the course!