January 21, 2019 January 21, 2019 10:00 PM PST 11:00 PM PST Mark your calendars  All GMAT Club Tests are free and open January 21st for celebrate Martin Luther King Jr.'s Birthday. January 22, 2019 January 22, 2019 10:00 PM PST 11:00 PM PST In case you didn’t notice, we recently held the 1st ever GMAT game show and it was awesome! See who won a full GMAT course, and register to the next one.
Author 
Message 
TAGS:

Hide Tags

Intern
Joined: 04 Sep 2009
Posts: 41

If x is a positive integer, is (x)(x + 2)(x + 4) div by 12?
[#permalink]
Show Tags
26 Nov 2012, 08:40
Question Stats:
61% (02:14) correct 39% (02:24) wrong based on 287 sessions
HideShow timer Statistics
If x is a positive integer, is (x)(x + 2)(x + 4) divisible by 12? (1) x^2 + 2x is a multiple of 3. (2) 3x is a multiple of 2.
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
Aeros "Why are you trying so hard to fit in when you were born to stand out?" "Do or do not. There is no 'try'..."




Math Expert
Joined: 02 Sep 2009
Posts: 52344

Re: If x is a positive integer, is (x)(x + 2)(x + 4) div by 12?
[#permalink]
Show Tags
26 Nov 2012, 08:58
If x is a positive integer, is (x)(x + 2)(x + 4) divisible by 12?Notice that no matter whether x is even or odd, out of x, x+2 and x+4 one must be a multiple of 3. So, we are basically asked to find whether (x)(x + 2)(x + 4) is divisible by 4. Now, if x=odd, then all three multiples are odd, thus (x)(x + 2)(x + 4) will be odd and not divisible by 4. If x=even, then (x)(x + 2)(x + 4)=even*even*even, thus it'll be divisible by 4. Therefore, the question boils down to find whether x is even. (1) x^2 + 2x is a multiple of 3 > if x=1=odd, then the answer is NO but if x=4, then the answer is YES. Not sufficient. (2) 3x is a multiple of 2. This statement implies that x=even. Sufficient. Answer: B. Hope it's clear.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics




Current Student
Joined: 26 Aug 2015
Posts: 33
Concentration: Strategy, Economics
GMAT 1: 570 Q40 V28 GMAT 2: 740 Q49 V41

If x is a positive integer, is (x)(x + 2)(x + 4) divisible by 12?
[#permalink]
Show Tags
14 Nov 2016, 10:17
If x is a positive integer, is (x)(x + 2)(x + 4) divisible by 12? (1) x2 + 2x is a multiple of 3. (2) 3x is a multiple of 2.
_________________
Send some kudos this way if I was helpful! !



Math Expert
Joined: 02 Sep 2009
Posts: 52344

Re: If x is a positive integer, is (x)(x + 2)(x + 4) div by 12?
[#permalink]
Show Tags
14 Nov 2016, 10:22



Current Student
Joined: 26 Aug 2015
Posts: 33
Concentration: Strategy, Economics
GMAT 1: 570 Q40 V28 GMAT 2: 740 Q49 V41

If x is a positive integer, is (x)(x + 2)(x + 4) div by 12?
[#permalink]
Show Tags
14 Nov 2016, 10:37
Bunuel wrote: Ilomelin wrote: If x is a positive integer, is (x)(x + 2)(x + 4) divisible by 12?
(1) x2 + 2x is a multiple of 3.
(2) 3x is a multiple of 2. Merging topics. Please refer to the solution above. Thank you Bunuel. Could you please explain me how do we know (other than plugging in numbers) that out of 3 consecutive odd or even integers, one must be divisible by 3? I understand why it works with consecutive integers, but why does it also apply to evenly spaced odds or evens? Thanks.
_________________
Send some kudos this way if I was helpful! !



Intern
Joined: 22 Jan 2017
Posts: 34

Re: If x is a positive integer, is (x)(x + 2)(x + 4) div by 12?
[#permalink]
Show Tags
18 Jul 2017, 08:43
Here's my interpretation, I'd actually love it if someone could tell me if I'm wrong here as I often make little mistakes in the theory on these questions.
Is (x)(x + 2)(x + 4) divisible by 12?
(1) x^2 + 2x is a multiple of 3. (2) 3x is a multiple of 2.
I immediately rephrase the 12 into (2^2)*(3). So we need to be divisible by 2 twice and 3 one. So contained in these three terms we need two multiples of 2 and one multiple of 3.
First observation is we either have consecutive even numbers or consecutive odd numbers. If it's the latter, the answer is definitely a 'no' and if it's the former it will be 'yes'.
Why? Here's a quick aside.
By the way this is not something you have time to do in a question so I would really recommend internalizing this concept as it is the backbone of a lot of problems on divisibility, remainders, and several other question families.
Think about the progression of integers (and this is a good thing to get comfortable with). We need two 2's and one 3 from these three integers. In general, we get a 2 every second integer and we get a 3 every third integer.
Now, if we have three odd numbers above what are we going to get? Three numbers that have ZERO 2's. That's what an odd number is: a number that is not divisible by 2. How many 3's will there be? Well we have three consecutive odd numbers, so three numbers that have a range of 4 and start on an odd number.
For instance:
3, 5, 7 or 13, 15, 17 or 21, 23, 25
We can see from these examples that that is always going to be one number that is a multiple of 3. Will it always be only one multiple of 3? Yes, only one. Why? Well, for the same reason we gave above. Multiples of 3 are going to occur every 3rd integer. Perhaps an easier way to understand this (it helps me) is to visualize it.
Consider three consecutive odd numbers:
5, 7, 9
From zero ascending that looks like this:
0 _ _ _ _ 5 _ 7 _ 9 _ ...
Where are the multiples of 3?
0 _ _ 3 _ 5 6 7 _ 9 _
Algebraically, we can say that for three numbers: x, x + 2, and x +4:
If (x) is a multiple of 3 then we have our multiple of three (and we can say the other two integers are not multiples of 3). If x is NOT a multiple of 3 then either (x + 1) or (x  1) is, but not both. If (x + 1) is, then so is (x + 4) which is the last of our three numbers. If (x + 1) and (x) are not, then (x  1) is and so is (x + 2) which is our second integer.
So we can see that for any three consecutive odd integers you are going to hit a multiple of 3 at some point in the three numbers.
Therefore in three consecutive odd integers we will have one multiple of 3 and no multiples of 2.
Now, what if we have three even numbers? Well, obviously there a bunch of 2's. What about 3's, how many will there be? Same logic as above. If x is a multiple of 3 we have one there. If x isn't then either (x + 1) or (x  1) is. If it's the former then x + 4 is a multiple of 3. If it's (x 1) then (x2) is a multiple of 3.
Ok so before we go to the statements we've proven (albeit laboriously) that the answer depends entirely on whether x is odd or even.
Statement 1: x^2 + 2x is a multiple of 3.
x*(x + 2) multiple of 3.
We don't know if they are even, this could be 6*8 or 3*5. INSUFFICIENT.
Statement 2: 3x is a multiple of 2.
So if 3x is a multiple of 2, then x must be even because 3*(?) = even then the (?) must be even. SUFFICIENT.
Another way to look at statement 2 is to rephrase the statement into:
3x = 2(i), where i is some integer x = (2/3)*(i)
Therefore for x to remain an integer i must be a multiple of 3. So x is a multiple of 2 and 3. Therefore x is (at a minimum) 6 and we have one 2, one 3 and then our second term in the question will provide the second 2 and we have all we need.



Manager
Joined: 15 Dec 2016
Posts: 102

If x is a positive integer, is (x)(x + 2)(x + 4) div by 12?
[#permalink]
Show Tags
27 Aug 2017, 16:37
Bunuel wrote: If x is a positive integer, is (x)(x + 2)(x + 4) divisible by 12?
Notice that no matter whether x is even or odd, out of x, x+2 and x+4 one must be a multiple of 3. So, we are basically asked to find whether (x)(x + 2)(x + 4) is divisible by 4. Now, if x=odd, then all three multiples are odd, thus (x)(x + 2)(x + 4) will be odd and not divisible by 4. If x=even, then (x)(x + 2)(x + 4)=even*even*even, thus it'll be divisible by 4.
Therefore, the question boils down to find whether x is even.
(1) x^2 + 2x is a multiple of 3 > if x=1=odd, then the answer is NO but if x=4, then the answer is YES. Not sufficient.
(2) 3x is a multiple of 2. This statement implies that x=even. Sufficient.
Answer: B.
Hope it's clear. Hi Bunuel  you mentioned the following : "Notice that no matter whether x is even or odd, out of x, x+2 and x+4 one must be a multiple of 3."  I AGREE So, we are basically asked to find whether (x)(x + 2)(x + 4) is divisible by 4. = QUESTION ON THIS STATEMENT Should it not be 2 out of the 3 is divisible by 4, i.e. we need to find whether x and (x+2) or (x+2) and (x+4) or (X+2) and (x+4) are divisible by 4 ....Given either x or (x+2) or (x+4) is going to be a multiple of 3 .....ONLY THE OTHER TWO have to be multiples of 4 instead ... Please let me know your thoughts



NonHuman User
Joined: 09 Sep 2013
Posts: 9462

Re: If x is a positive integer, is (x)(x + 2)(x + 4) div by 12?
[#permalink]
Show Tags
18 Sep 2018, 21:21
Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up  doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
GMAT Books  GMAT Club Tests  Best Prices on GMAT Courses  GMAT Mobile App  Math Resources  Verbal Resources




Re: If x is a positive integer, is (x)(x + 2)(x + 4) div by 12? &nbs
[#permalink]
18 Sep 2018, 21:21






