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Manager  Joined: 22 Mar 2011
Posts: 52
If a and b are both single-digit positive integers, is a + b  [#permalink]

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12 00:00

Difficulty:   55% (hard)

Question Stats: 66% (02:06) correct 34% (02:02) wrong based on 332 sessions

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If a and b are both single-digit positive integers, is a + b a multiple of 3?

(1) The two-digit number "ab" (where a is in the tens place and b is in the ones place) is a multiple of 3.

(2) a – 2b is a multiple of 3.

If we take a scenario for part 1

A = 9 B = 3
A x B = 27
A + B = 9 (Divisible by 3)

A = 4 B = 3
A x B = 12
A + B = 7 (Not Divisible by 3)

So it should be insufficient. Please help.
Senior Manager  Joined: 24 Mar 2011
Posts: 341
Location: Texas
Re: Multiples of 3  [#permalink]

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2
i think you misunderstood it.

what it is saying you is
The two-digit number "ab" (where a is in the tens place and b is in the ones place) is a multiple of 3.

meaning integer like 10 where a = 1 and b=0; 11 where a =1, b=1 ; 12 where a = 1 and b =2
(note that it reads 2 digit number 'ab' and is not a * b)

so basically all the numbers that satisfy condition 1 will be 12, 15, 18, 21, 24, 27, 30, 33, 36, 39....and so on
notice that are multiple of 3.

now a+b for all these numbers = 3, 6, 9, 3, 6, 9, 3, 6 follow the patter of 3, 6 , 9 and are multiple of 3.
Current Student Joined: 26 May 2005
Posts: 495
Re: Multiples of 3  [#permalink]

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1
melguy wrote:
If a and b are both single-digit positive integers, is a + b a multiple of 3?

(1) The two-digit number "ab" (where a is in the tens place and b is in the ones place) is a multiple of 3.

(2) a – 2b is a multiple of 3.

If we take a scenario for part 1

A = 9 B = 3
A x B = 27
A + B = 9 (Divisible by 3)

A = 4 B = 3
A x B = 12
A + B = 7 (Not Divisible by 3)

So it should be insufficient. Please help.

Question asks if a+b is divisible by 3

1. pick up the numbers 12, 15,18, 21 24, 30 all are multiple of 3 and a+b ( 1+2,1+5,1+9....) is a multiple of 3 - hence sufficient

2. if a is a multiple of 3 and b is a multiple of 3 then a-b is also a multiple of 3
pick up numbers for a-2b, remember this should be a multiple of 3., thus a is multiple of 3 and 2b is a multiple of 3
a = 9 b = 3 then a-2b is 3 ..multiple of 3
a= 24 b= 6 a-2b = 12 multiple of 3..........sufficient

hence D
Current Student Joined: 08 Jan 2009
Posts: 303
GMAT 1: 770 Q50 V46 Re: Multiples of 3  [#permalink]

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1
1) is also the check for 'division by 3'.

I.e. a number is a multiple of 3 if its digits added together are divisible by 3. Therefore, if ab is divisible by 3, then a+b must be divisible by 3.
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Joined: 16 Nov 2010
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Location: United States (IN)
Concentration: Strategy, Technology
Re: Multiples of 3  [#permalink]

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(1)

Clearly Sufficient as "ab" is a multiple of 3 when a+b is so

You should consider AB as 93, instead of A * B = 27

(2)

a - 2b = 3k, where k is an integer

=> a = 3k + 2b

=> ab is in a format "3k+2bb"

So a + b = 3k + 2b + b = 3(k+b), a multiple of 3

=> ab is a multiple of 3

Sufficient

Answer - D
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Math Expert V
Joined: 02 Sep 2009
Posts: 55271
If a and b are both single-digit positive integers, is a + b a multipl  [#permalink]

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Tough and Tricky questions: Multiples.

If a and b are both single-digit positive integers, is a + b a multiple of 3?

(1) The two-digit number "ab" (where a is in the tens place and b is in the ones place) is a multiple of 3.
(2) a – 2b is a multiple of 3.
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Intern  Joined: 27 Jul 2012
Posts: 25
Re: If a and b are both single-digit positive integers, is a + b a multipl  [#permalink]

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2
1
Answer is D, both are sufficient.

1) for any number to be a multiple of 3, the sum of its digits needs to be a multiple of 3.
So therefore, if given ab is a multiple of 3, then a + b is also a multiple of 3.

2) a -2b is a multiple of 3. If we add to this a multiple of 3, the resultant sum should also be a multiple of 3.
a-2b + 3b(multiple of 3) = a+b which by virtue of being a sum of 2 multiple of 3, is also a multiple.

Thus each statement alone is sufficient
Manager  S
Joined: 22 Jan 2014
Posts: 173
WE: Project Management (Computer Hardware)
Re: If a and b are both single-digit positive integers, is a + b a multipl  [#permalink]

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3
Bunuel wrote:

Tough and Tricky questions: Multiples.

If a and b are both single-digit positive integers, is a + b a multiple of 3?

(1) The two-digit number "ab" (where a is in the tens place and b is in the ones place) is a multiple of 3.
(2) a – 2b is a multiple of 3.

D.

1) ab is a multiple of 3.
By basic rule of division, a number is div by 3 if the sum of its digits is div by 3.
Reason: 10a+b mod 3 = 0
=> (a+b) mod 3 = 0 [since 10 mod 3 = 1]
so sufficient.

2) a-2b = 3k
a = 3k+2b
=> a+b = 3k+2b+b = 3(k+b) which is div by 3
so sufficient.
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Intern  B
Joined: 27 Dec 2015
Posts: 29
Re: If a and b are both single-digit positive integers, is a + b a multipl  [#permalink]

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I know this is an older post, but can someone clarify how it is we go from => a = 3k + 2b to a + b = 3k + 2b + b = 3(k+b) for statement 2?

thanks Math Expert V
Joined: 02 Sep 2009
Posts: 55271
Re: If a and b are both single-digit positive integers, is a + b a multipl  [#permalink]

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mogarza08 wrote:
I know this is an older post, but can someone clarify how it is we go from => a = 3k + 2b to a + b = 3k + 2b + b = 3(k+b) for statement 2?

thanks a = 3k + 2b;

Add b to both sides: a + b = (3k + 2b) + b;

a + b = 3k + 3b;

a + b = 3(k + 3).
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Intern  B
Joined: 27 Dec 2015
Posts: 29
Re: If a and b are both single-digit positive integers, is a + b a multipl  [#permalink]

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makes perfect sense now, thank you!!
Math Revolution GMAT Instructor V
Joined: 16 Aug 2015
Posts: 7372
GMAT 1: 760 Q51 V42 GPA: 3.82
Re: If a and b are both single-digit positive integers, is a + b a multipl  [#permalink]

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Before solving the question, you need some knowledge. “The remainder of a certain positive integer divided by 3 and 9 is same as the sum of every digit place of n divided by 3 and 9.
There are 2 variables in the original condition. In order to match the number of variables to the number of equations, we need 2 equations. Since the condition 1) and the condition 2) each has 1 equation, there is high chance that C is the correct answer.
Using the condition 1) and the condition 2) at the same time:
Condition 1) – a multiple of ab=3 is a multiple of a+b=3. Hence, the answer is yes and the condition is sufficient.
Condition 2) – If a+b is a multiple of a-2b=3, then, it becomes a multiple of a-2b+3b=3 plus 3b and a multiple of a+b=3 plus 3b=3(multiple+b)=3. Hence, the answer is yes and the condition is sufficient. Thus, the correct answer is D.
This is 50-51 level question. Please remember the common mistake type 4(B). If the answer is too easily A or B, please consider D.

- For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
Attachments variable approach's answer probability.jpg [ 219.74 KiB | Viewed 4284 times ]

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Intern  B
Joined: 03 Sep 2017
Posts: 3
Re: If a and b are both single-digit positive integers, is a + b a multipl  [#permalink]

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1) two digit integer ab is a multiple of 3 <=> (a+b) = 3k => clearly sufficient
2) a-2b =3k => a-2b+3b is also a multiple of 3 => a+b = 3k => sufficient
=> D
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Re: If a and b are both single-digit positive integers, is a + b a multipl  [#permalink]

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_________________ Re: If a and b are both single-digit positive integers, is a + b a multipl   [#permalink] 24 Nov 2018, 07:35
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