Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 44586

The integers a, b, c, and d can each be equal to 0, 1, 2, or 3, indepe [#permalink]
Show Tags
24 Apr 2015, 03:07
2
This post received KUDOS
Expert's post
2
This post was BOOKMARKED
Question Stats:
59% (02:21) correct 41% (03:25) wrong based on 99 sessions
HideShow timer Statistics
The integers a, b, c, and d can each be equal to 0, 1, 2, or 3, independently. What is the value of (a + 1)(b + 1)(c + 1)(d + 1)? (1) a + 4b + 16c + 64d = 165 (2) 64a + 16b + 4c + d = 90
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
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



Retired Moderator
Joined: 06 Jul 2014
Posts: 1266
Location: Ukraine
Concentration: Entrepreneurship, Technology
GMAT 1: 660 Q48 V33 GMAT 2: 740 Q50 V40

Re: The integers a, b, c, and d can each be equal to 0, 1, 2, or 3, indepe [#permalink]
Show Tags
24 Apr 2015, 08:46
1
This post received KUDOS
2
This post was BOOKMARKED
Bunuel wrote: The integers a, b, c, and d can each be equal to 0, 1, 2, or 3, independently. What is the value of (a + 1)(b + 1)(c + 1)(d + 1)?
(1) a + 4b + 16c + 64d = 165
(2) 64a + 16b + 4c + d = 90 Picking numbers approach: 1) let's start from biggest numbers: max \(d\) can be \(2\) and we have \(2*64 = 128\); max \(c\) can be \(2\) so \(2*16=32\); \(128+32 = 160\) and we need \(5\) so \(b=1\) and \(a=1\). \(1*1+4*1+16*2+64*2=165\) We've found first variant and should seek for another possible scenarios: \(d = 1\) so \(1*64 = 64\) and for \(c\) we pick \(3\) so \(3 * 16 = 48\); \(64+48 =112\) and we need \(53\) and we can't make such number from \(4b\) and \(a\), because maximum can be \(4*3+1*3 = 15\) Sufficient 2) Max \(a\) can be \(1\) so \(1*64\); max \(b\) can \(1\) so \(1*16= 16\) and we have \(80\) and need \(10\) more so \(c =2\) and \(d = 2\). \(64*1+16*1+4*2+1*2=90\) And if we try another variants we can see that we don't have other variants. Sufficient Answer is D.
_________________
Simple way to always control time during the quant part. How to solve main idea questions without full understanding of RC. 660 (Q48, V33)  unpleasant surprise 740 (Q50, V40, IR3)  antidebrief



Director
Joined: 07 Aug 2011
Posts: 568
Concentration: International Business, Technology

Re: The integers a, b, c, and d can each be equal to 0, 1, 2, or 3, indepe [#permalink]
Show Tags
25 Apr 2015, 01:58
1
This post received KUDOS
Bunuel wrote: The integers a, b, c, and d can each be equal to 0, 1, 2, or 3, independently. What is the value of (a + 1)(b + 1)(c + 1)(d + 1)?
(1) a + 4b + 16c + 64d = 165
(2) 64a + 16b + 4c + d = 90 (1) a + 4b + 16c + 64d = 165 a=1,b=1, c=2,d=2 (2) 64a + 16b + 4c + d = 90 a=1, b=1,c=2,d=2 Answer D.
_________________
Thanks, Lucky
_______________________________________________________ Kindly press the to appreciate my post !!



Manager
Joined: 24 Jan 2015
Posts: 67
GPA: 4
WE: Consulting (Pharmaceuticals and Biotech)

Re: The integers a, b, c, and d can each be equal to 0, 1, 2, or 3, indepe [#permalink]
Show Tags
25 Apr 2015, 20:44
1
This post received KUDOS
The integers a, b, c, and d can each be equal to 0, 1, 2, or 3, independently. What is the value of (a + 1)(b + 1)(c + 1)(d + 1)?
(1) a + 4b + 16c + 64d = 165
From Statement 1 , we know that the sum of integers is equal to an ODD number (165) Hence, one of the 4 integers has to be an odd number ==> Since 4b, 16c and 64d always result in even, a must take an odd value i.e 1 or 3 only
a = {1,3}
a + 4b + 16c + 64d = 165 > Lets start with 'd' . The possible values for d are 0,1,2 and 3.
d can take only 0,1 & 2 because d=3 gives a value >165. And since there is no way we can achieve 165 by substituting values 0 & 1 , d must be 2. ==> d = 2
a + 4b + 16c = 37 > Lets go with 'c' now. The possible values for c are 0,1,2 and 3.
c can take only 0,1 & 2 because c=3 gives a value >37. And since there is no way we can achieve 37 by substituting values 0 & 1 , c must be 2. ==> c = 2
a + 4b = 5  > We know a must be odd i.e 1 or 3 only and possible values for b are 0,1,2 and 3.
Substituting a =3 , we get 4b = 1 (not possible because b is an integer)
Therefore, a = 1 and b = 1
Statement 1 is sufficient
(2) 64a + 16b + 4c + d = 90
From Statement 2 , we know that the sum of integers is equal to an EVEN number (90) Hence, all 4 integers have to be an even ==> Since 64a, 16b and 4c always result in even, d must take an even value i.e 0 or 2 only
a = {0,2}
64a + 16b + 4c + d = 90 > Lets start with 'a' . The possible values for a are 0,1,2 and 3.
a can take only 0 & 1 because a= 2 or 3 gives a value >90. And since there is no way we can achieve 90 by substituting value 0 , a must be 1. ==> a = 1
16b + 4c + d = 26 > Lets go with 'b' now. The possible values for b are 0,1,2 and 3.
b can take only 0 & 1 because b = 2 or 3 gives a value >26. And since there is no way we can achieve 26 by substituting value 0 , b must be 1. ==> c = 1
4c + d = 10  > We know d must be even i.e 0 or 2 only and possible values for c are 0,1,2 and 3.
Substituting d =0 , we get 4c = 10 (not possible because c is an integer)
Therefore, d = 2 and c = 2
Statement 2 is sufficient
Answer is (D)



Manager
Joined: 15 May 2014
Posts: 65

Re: The integers a, b, c, and d can each be equal to 0, 1, 2, or 3, indepe [#permalink]
Show Tags
25 Apr 2015, 23:51
2
This post received KUDOS
The integers a, b, c, and d can each be equal to 0, 1, 2, or 3, independently (a + 1)(b + 1)(c + 1)(d + 1) = ?
Statement (1): a + 4b + 16c + 64d = 165 a + 4(b + 4c + 16d) = 165 odd + even = odd a = 1 or 3 if a = 1, 164 is divisible by 4, if a = 3, 162 is not divisible by 4 > a = 1
b + 4c + 16d = 41 b + 4(c + 4d) = 41 odd + even = odd b = 1 or 3 if b = 1, 40 is divisible by 4, if b = 3, 38 is not divisible by 4, > b = 1
c + 4d = 10 even + even = even c = 0 or 2 if c = 0, 10 is not divisible by 4, if c = 2, 8 is divisible by 4, > c = 2
4d = 8 > d = 2 Sufficient
Statement (2) 64a + 16b + 4c + d = 90 4(16a + 4b + c) + d = 90 even + even = 90 d = 0 or 2 if d = 0, 90 is not divisible by 4, if d = 2, 88 is divisible by 4, > d = 2
16a + 4b + c = 22 4(4a + b) + c = 22 even + even = even c = 0 or 2 if c = 0, 22 is not divisible by 4 if c = 2, 20 is divisible by 4 > c = 2
4a + b = 5 even + odd = odd b = 1 or 3 if b = 1, 4 is divisible by 4 b = 3, 2 is not divisible by 4 > b = 1
4a = 4 > a = 1 Sufficient
Answer D



Math Expert
Joined: 02 Sep 2009
Posts: 44586

Re: The integers a, b, c, and d can each be equal to 0, 1, 2, or 3, indepe [#permalink]
Show Tags
27 Apr 2015, 01:56
Bunuel wrote: The integers a, b, c, and d can each be equal to 0, 1, 2, or 3, independently. What is the value of (a + 1)(b + 1)(c + 1)(d + 1)?
(1) a + 4b + 16c + 64d = 165
(2) 64a + 16b + 4c + d = 90 MANHATTAN GMAT OFFICIAL SOLUTION:After you note the constraints on a, b, c, and d, focus on the question. Because multiplying the four variables (or, rather, 1 plus each of those variables) together can produce a ton of possible outcomes, there’s not much use in listing all those outcomes out and looking for patterns. Instead, just rephrase to a simple “What are the values of a, b, c, and d?” (Technically, that’s an oversimplification, but it will do for now—and when you rephrase, don’t ever forget completely about the original phrasing of the question.) Statement (1): SUFFICIENT. Surprisingly, this equation provides enough information to find the values of all four variables. A good way to see why is to consider the following variation on the problem. Imagine that you have three new variables (x, y, and z), and that these variables can take on any integer value from 0 to 9, inclusive. Can you solve for all three variables from the following equation? 100x + 10y + z = 243 Yes, you could! x = 2, y = 4, and z = 3. We’re all used to the idea with typical base10 numbers that you can express any positive integer uniquely as a sum of 09 units, 09 tens, 09 hundreds, etc. In other words, you can express any integer uniquely as a sum of powers of ten, if you ensure that you never take more than 9 copies of any power of ten. (In other words, you always take a digit number of copies—for instance, in 243, you take 2 hundreds, 4 tens, and 3 units.) You can do the same thing with any base, as long as you adjust the possible number of copies down. For instance, to express a positive integer as a unique sum of powers of 4, you must limit yourself to taking no more than 3 copies of any power. (In the same way, you can’t take 10 hundreds when you write out a number with digits: the way to do so is to take 1 thousand.) If you knew all this in advance, great! What if you didn’t? You could get there by fiddling. You want to get to 165. Build up from the biggest power. Make d equal to 2, so 64d = 128. a + 4b + 16c + 128= 165 Okay, now you have to get to 165 – 128 = 37. Make c = 2, so 16c = 32. a + 4b + 32 = 37 Finally, make a = 1 and b = 1, and you’re there. There’s no other way to get to 165, as you can confirm by trying other possible values of the four variables. Thus, you can find a unique value of the expression in the question. Statement (2): SUFFICIENT. For the same reasons, you can solve uniquely for the values of the four variables. Again, they are a = 1, b = 1, c = 2, and d = 2. The correct answer is D.
_________________
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



NonHuman User
Joined: 09 Sep 2013
Posts: 6646

Re: The integers a, b, c, and d can each be equal to 0, 1, 2, or 3, indepe [#permalink]
Show Tags
30 Mar 2018, 21:40
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: The integers a, b, c, and d can each be equal to 0, 1, 2, or 3, indepe
[#permalink]
30 Mar 2018, 21:40






