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If a and b are both singledigit positive integers, is a + b [#permalink]
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28 May 2011, 19:34
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If a and b are both singledigit positive integers, is a + b a multiple of 3? (1) The twodigit 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.
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Re: Multiples of 3 [#permalink]
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28 May 2011, 20:33
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i think you misunderstood it.
what it is saying you is The twodigit 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.



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Re: Multiples of 3 [#permalink]
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28 May 2011, 21:59
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melguy wrote: If a and b are both singledigit positive integers, is a + b a multiple of 3?
(1) The twodigit 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 ab is also a multiple of 3 pick up numbers for a2b, 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 a2b is 3 ..multiple of 3 a= 24 b= 6 a2b = 12 multiple of 3..........sufficient hence D



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Re: Multiples of 3 [#permalink]
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29 May 2011, 02:12
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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Re: Multiples of 3 [#permalink]
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29 May 2011, 02:16
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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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If a and b are both singledigit positive integers, is a + b a multipl [#permalink]
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24 Oct 2014, 09:21



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Re: If a and b are both singledigit positive integers, is a + b a multipl [#permalink]
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24 Oct 2014, 16:44
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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. a2b + 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



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Re: If a and b are both singledigit positive integers, is a + b a multipl [#permalink]
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25 Oct 2014, 05:14
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Bunuel wrote: Tough and Tricky questions: Multiples. If a and b are both singledigit positive integers, is a + b a multiple of 3? (1) The twodigit 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) a2b = 3k a = 3k+2b => a+b = 3k+2b+b = 3(k+b) which is div by 3 so sufficient.
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Re: If a and b are both singledigit positive integers, is a + b a multipl [#permalink]
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13 Jun 2016, 11:51
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



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Re: If a and b are both singledigit positive integers, is a + b a multipl [#permalink]
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13 Jun 2016, 11:58



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Re: If a and b are both singledigit positive integers, is a + b a multipl [#permalink]
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13 Jun 2016, 12:09
makes perfect sense now, thank you!!



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Re: If a and b are both singledigit positive integers, is a + b a multipl [#permalink]
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14 Jun 2016, 09:26
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 a2b=3, then, it becomes a multiple of a2b+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 5051 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.
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Re: If a and b are both singledigit positive integers, is a + b a multipl [#permalink]
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20 Nov 2017, 12:13
1) two digit integer ab is a multiple of 3 <=> (a+b) = 3k => clearly sufficient 2) a2b =3k => a2b+3b is also a multiple of 3 => a+b = 3k => sufficient => D




Re: If a and b are both singledigit positive integers, is a + b a multipl
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20 Nov 2017, 12:13






