Q What is the remainder when the positive integer x is

Question

Q 
What is the remainder when the positive integer x is divided by 7

1) x-1 is divisible by 7 
2) x + 1 is divisible by 7 
Ø Statement 1 alone is sufficient but statement 2 alone is not sufficient 
Ø Statement 2 alone is sufficient but statement 1 alone is not sufficient 
Ø Both statements together are sufficient, but neither statement alone is sufficient 
Ø Each statement alone is sufficient 
Ø Statements 1 and 2 together are not sufficient

Q 
What is the remainder when x ^ 4 – y ^ 4 is divided by 3

(1) When x-y is divided by 3, remainder is 0 
(2) When x+y is divided by 3, remainder is 2 
Ø Statement 1 alone is sufficient but statement 2 alone is not sufficient 
Ø Statement 2 alone is sufficient but statement 1 alone is not sufficient 
Ø Both statements together are sufficient, but neither statement alone is sufficient 
Ø Each statement alone is sufficient 
Ø Statements 1 and 2 together are not sufficient

If x is a positive integer is x divisible by 7

1) X+29 is divisible by 10 
2) x + 10 is divisible by 29 
Ø Statement 1 alone is sufficient but statement 2 alone is not sufficient 
Ø Statement 2 alone is sufficient but statement 1 alone is not sufficient 
Ø Both statements together are sufficient, but neither statement alone is sufficient 
Ø Each statement alone is sufficient 
Ø Statements 1 and 2 together are not sufficient 
Can you please explain what the concept behind the above mentioned questions is? I am not looking at a solution which includes picking up a number and substituting it in the equation. The technique of inputting a number is valid for only very few types of question. Would really like to understand the trick and the concept behind these types of questions?

Disclaimer: - 
I do not know the source of this question, so please do not ask it from me. A friend of mine who just wrote his GMAT and got a cool 750 gave me the pdf files containing some challenging questions. I have the answers to these questions but not the official explanations.

buffdaddy · Answer

we can position the question like  x = 7k + R  and we need to find R

from 1) we can deduce that  x-1 = 7n  .
given this we can solve for R by setting k=n. i.e 7k + R = 7k + 1. Therefore R = 1 ---> Suff.

similarly for 2) we can decude R = -1, which actually means that R=6

answer = D

lets fist simplyfy  {x}^{4} - {y}^{4}  into simpler forms that will looks similar to the data options.

{x}^{4} - {y}^{4}  =  ({x}^{2} - {y}^{2})({x}^{2} + {y}^{2}) 
which in turn equals  (x+y)(x-y)({x}^{2} + {y}^{2})

so we need to find out if this term is divisible by 3.

1) says that (x-y) is divisible by 3. Therefore  {x}^{4} - {y}^{4}  is divisible by 3 --> suff

2) says that  \frac{(x+y)}{3}  yields a reminder of 2. -->insuff

Ans = A

this is a toughie +1 for posting it, I am keen to know what others come up with

ventivish · Answer

Hi, I think the answer to this question is E. I used a combination of picking numbers and divisibility rules. This may or may not be helpful. I am sure there must be a shorter way of approaching this problem.Unfortunately this was all I could think of

Stat 1 x+29 is divisible by 10 The units digit of x+29 must end in 0, therefore the units digit of x must be 1 x could be 1,11,which are not divisible by 7 and 21 which is. Insuff Stat 2 x+10 is divisible by 29, The units digits of numbers divisible by 29 range from 9,8,7,6,5,4,3,2 Therefore the units digit of x could be any of these numbers and if x+10 = 87 which is divisible by 29 and x =7 which is divisible by 7 or x+10=58 which is divisible by 29 and x=48 which is not divisible by 7. Insufficient Together stat 1 and 2 x+29 is divisible by 10 and x+10 is divisible by 29 Therefore , x could be 251,541,831, 1121 which is not divisible by 7 or x could be 1711 which is divisible by 7 so insuff.

RyanDe680 · Answer

This might sound dumb, but without doing any math, couldn't I just say that the answer is D? You can find the remainder when (x+1) or (x-1) is divided by 7, even if X is 6 or 8 and the remainder is zero, correct? :oops:

sirlogic · Answer

Whoa. How did you find this particular number out from just x+29 and x+10?? I would be interested in knowing your method.

FN · Answer

If x is a positive integer is x divisible by 7

1) X+29 is divisible by 10 
2) x + 10 is divisible by 29

if x=1, then x+29=30/10 1/7 is not even divisible by 7...
if x=21, sum=50 which is divisible by 10, however 21/7 is even divisible..so Insuff

2) x+10 is divisible by 29..which means if x=19 then x/7 is not even divisible..

lets look at 29*2=58..ok so pick x=48 48/7 is not even divisible by 7..

lets look at 29*3=87, so make x=77/7 is divisible by 7..

lets look at 1&2 X is sume number that has a unit digit 1..
aha 29*9 gives unit digit 1..so 29*9=201 201-10=191 OK so x=191

191+29=220 divisible by 10..

191+10=201 divisible by 29..

so lets check if 191 is divisible by 7..191/7 is not even divisible

therefore I am going to say C is my ans..

zonk · Answer

I don't think your test for C generalizes:

29*59 =1711
x = 1711 - 10 = 1701
1701/7 = 243

To re-write your methodology into a generalization:
x = 29*(10a+9)-10 = 290a + 251 where a is any non-negative integer

290a+251 is divisible by 7 for all a = 5b+7 where b is any non-negative integer

Therefore, E.

FN · Answer

yeah u r right..

on exam day i would have picked C..i mean how can you quickly check for all multiples of 29 that have 1 as unit digit..

zonk · Answer

I would bet this is a manhattan gmat CAT problem and not a gmatprep problem...

jallenmorris · Answer

251,541,831, 1121 is not a single number, this is a list of 4 separate numbers.

brokerbevo · Answer

Isn't the first question impossible? When you take the 2 statements together, it states a number is divisible by 7 and when you add 2 to it, it still divisible by 7. For example, lets say x is 6, if you add 1 (x + 1) you will get 7 (which is divisible by 7), but if you subtract 1 (x-1), you get 5, and that is not divisible by 7. In other words, there is no value for x that satisfies both statement (1) and (2)---where if you take a number and whether you add 1 or subtract 1 from it, it will still be divisible by 7. Thus, this question is not a possible GMAT question. Another way of looking at it, is that for statement (1) you get a remainder of 1 and for statement 2, you get a remainder of 6 --> this is not possible, a single number can not have a remainder of one value, and then have a different remainder when divided by the same number. Please feel free to comment as I am 90% sure on this.

jallenmorris · Answer

brokerbevo,

You're taking both statements together when you don't need to. The question is what is the remainder when x is divided by 7. And then it gives you the statements. Its easy to see that each statement, separately, allows you to easily find the remainder when x is divided by 7. You don't have to go to each statement, but even if you did, the answer would then be E because the statements together don't allow you to answer the question.

I think this goes back to our conversation of the statements cannot contradict each other. Here is an example of when the statements can contradict each other and the question is perfectly acceptable. As for it being a GMAT question? I have no idea. I don't know the source of this particular question.

Anyone else know the source of this question?

brokerbevo · Answer

No, I know. I do realize that when you view each statement in isolation you do in fact get a single remainder for each: (1) remainder = 1 and (2) remainder = 6. However, there is no number in which if you add 1 or subtract 1 from it, and it will be divisible by 7. It is impossible so thus the GMAT will not give you a question like this (the statements are not true).

jallenmorris · Answer

But the point of Data Sufficiency is not to find out if 2 statements are sufficient, we have to take each statement individually first. Then, and ONLY THEN, do we consider the 2 statements together. I think if neither statment is sufficient by itself, then the statements are sure to always be consistent. It seems like you jump to consider the statements together.

With Data Sufficiency, our ability to analyze the amount of information available as well as the relevance of that information. First we look to the stem and determine what information is missing. Then we look to statement #1, figure out if it has the missing information, or other information in order for us to deduce the missing information. Then we look to #2 and do the same. ONLY if neither statement is sufficient do we consider the two statements together. I think it is entirely possible that 2 statements are inconsistent, but one of them is sufficient to answer the question so we never need to consider them together.

You know I think even after we consider the statements together they can be inconsistent, because then we would just answer E.

brokerbevo · Answer

Ahh, I see what you are saying. No, what I was saying is that I did take each question individually first: I found a remainder of 1 for statement (1) and a remainder of 6 for statement (2). However, I'm stating that the entire question in invalid because no single number, when you add 1 to OR subtract 1 from it, will be divisible by 7-- its impossible from a GMAT standpoint, the GMAT only deals with real numbers. For example, lets say you have the following numbers:

6: +1 = 7 AND -1 = 5 Both answers are not divisible by 7
8: +1 = 9 AND -1 = 7 Both answers are not divisible by 7
22: +1 = 23 AND -1 = 21 Both answers not divisible by 7

....no matter what number you try, when you subtract 1 and add 1, one or the other will not be divisible by 7. The statements do not agree with each other (in other words, they are contradictory), therefore the question will not be seen on the GMAT. A good way to tell if it is contradictory is to see if you can find a single value that satisfies both statements (of course, this is after you have done all your work and came up with an answer), if you can't, either the question is not a GMAT question, and you wouldn't see it on the test anyway, or you have made a math error. Can you think of a single value x, where x + 1 AND x - 1 are both divisible by 7?

jallenmorris · Answer

I agree with the method you identify to determine if the statements are contradictory, but where in the GMAT rules or instructions, or anything published by GMAC does it say that you will always find a number that will satisfy both statements?

Also, the GMAT deals with more than real numbers. Try  \sqrt{3} .

brokerbevo · Answer

Well, the definition of contradictory, in this context, is just that: if you can't find a single value (or more values) common among the answer choices, they are contradictory. That is the definition of contradictory when it comes to numbers.

If statement (1) says x < 0 and statement (2) says 5 < x, these statements are contradictory--meaning no single value (or more) can satisfy both statements. You will never see this on the GMAT because they contradict.

Now, if statement (1) said x^2 = 16 and statement (2) said x < 0 and you came to the conclusion that (1) x = 4 and (2) x is negative, you know you would have done some math wrong because how can a negative number equal 4? As silly as it sounds, it obviously doesn't make sense. In other words, a single value does not exist between the statements. However, statement (1) should have been calculated to be: x = +/- 4 and (2) would still state that x is negative. Now we have a value for x for both statements: -4.

Its not a matter of whether the statements are alone sufficient, only sufficient together, or not sufficient at all, its a matter of whether its a gmat question or not. In fact, its a moot point because you wouldn't see this on the GMAT anyway; however, it is a good double-check to make sure your math is correct.

IanStewart · Answer

brokerbevo is entirely correct. The statements are contradictory; you'd never see this question on a real GMAT. Clearly D is a good answer here, but logically E is just as good- there's no way to work out the remainder with both statements, since the situation is impossible. The GMAT can't have contradictory statements, because this will make two answers equally valid, and a GMAT question can only have one correct answer.

brokerbevo · Answer

Well, I think jallenmorris and I are both correct but we are just arguing about 2 different things-- just a miscommunication. He agrees that the statements can't contradict, but he was just stating the process of how to go about validating the statements: first you see if statement 1 is sufficient, then 2, then both, then none of them. I was just stating as a double-check, when all is said and done, that the outcome of your work from each statement must agree with each other. So, 2 completely different arguments. DS statements are confusing so its hard to communicate with others some of your techniques and whatnot. :-D

Q What is the remainder when the positive integer x is

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards