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# If a and b are positive integers, is 10^a + b divisible by 3

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If a and b are positive integers, is 10^a + b divisible by 3  [#permalink]

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Updated on: 12 Oct 2013, 09:15
3
14
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Difficulty:

55% (hard)

Question Stats:

67% (02:10) correct 33% (02:34) wrong based on 211 sessions

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If a and b are positive integers, is 10^a + b divisible by 3?

(1) b/2 is an odd integer.

(2) The remainder of b/10 is b.

Originally posted by study on 06 Nov 2008, 10:53.
Last edited by Bunuel on 12 Oct 2013, 09:15, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
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Re: If a and b are positive integers, is 10^a + b divisible by  [#permalink]

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12 Oct 2013, 07:36
10
study wrote:
If $$a$$ and $$b$$ are positive integers, is $$10^a + b$$ divisible by 3?

1. $$\frac{b}{2}$$ is an odd integer
2. the remainder of $$\frac{b}{10}$$ is $$b$$

what positive integer divided by 10 gives a remainder of itself?

Hi all, let me try to explain this one.

So we have 10^a + b /3 what is the remainder?

First of all note that you are told that a,b are positive integers. THEY ALWAYS TELL YOU THIS FOR SOME REASON, DON'T IGNORE IT

So we have that 10^a = (10,100 etc...)

Now, what is the divisibility rule for multiples of 3? That's right the digits of the number must add to a multiple of 3
We already have the 1 on the 10^a. Let's see what we can get out of 'b'

Statement 1: b/2 is odd integer. So b could be (2,6,10,14 etc...)
If b is 2 then remainder is 0
If b is 6 then remainder is 1
If b is 10 then remainder is 2
Not good enough

Statement 2: b/10 remainder is b. Now wait a second. What this is telling us is that b<10. But that could be (1,2,3,4,5 etc...)
This is stil not enough

Statement (1) and (2) - Now going back to statement 1. We still have 2 or 6 as possible answer giving different remainders

Hence (E) is the correct answer

Kudos if you like!
##### General Discussion
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06 Nov 2008, 11:08
1
for a number to be divisible by 3, sum of the digits have to be divisible by 3.
So I'd try looking to get at the sum of digits from $$10^a + b$$
$$10^a$$ gives us just 1+how many ever 0s. so the options for sum of the digits of b ought to be 2, 4, 8 etc..

From (1) we are told b/2 is odd.. Take for example b = 6, so b/2 = 3. But then sum of digits yields 1+3, which is not divisible by 3. Alternatively, take b=10, b/2 = 5. Then sum of digits yields 5+1 = 6, which is divisible by 3. Since we have conflicting results from (1), it is not sufficient by itself. Cross out A and D.

(2) says remainder of b/10 is b. That means, b < 10. We'll get conflicting results if we were to chose b=5 or b = 4. So (2) alone is not sufficient enough. Cross out B.

(1) and (2) together, says b/2 is odd and b < 10, it gives only choice for b as 6. that is conclusive enough. $$10^a + 6$$ is not divisible by 3. So my pick for the answer is C
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06 Nov 2008, 11:36
[quote="study"]If a and b are positive integers, is 10^a + b divisible by 3?

1. b/2 is an odd integer
2. the remainder of b/10 is b

when 10^a is devided by 3 the remainder is always 1 thus the question asks is 1+b devisable by 3
b can be 2,5,8,11,14..etc

from 1

b is even .....insuff

from 2

b is is less than 10 (0,1,2................9).........insuff

both

2,4,6,8 are all even and less than 10.........ONLY 6 WHEN DEVIDED BY 2 GIVES AN ODD

SUFF

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06 Nov 2008, 11:58
1
agree with C..

stmnt 2 says B<10

stmnt 1 says B/2=odd

together we know b=6
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06 Nov 2008, 12:04
2
study wrote:
If $$a$$ and $$b$$ are positive integers, is $$10^a + b$$ divisible by 3?

1. $$\frac{b}{2}$$ is an odd integer
2. the remainder of $$\frac{b}{10}$$ is $$b$$

what positive integer divided by itself gives a remainder of itself?

It should be E because b could be 2 or 6.

If b is either 2, 10^a + b is divisible by 3. But if b is 6, then 10^a + b is not divisible by 3.
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06 Nov 2008, 12:14
1
yup ur right..i overlooked 2/2=1
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06 Nov 2008, 14:38
thanks for pointing out a silly mistake

stand corrected. Somehow on DS, I gravitate towards C. Need to keep that in check on the real test.
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06 Nov 2008, 20:49
Yeah its E

b/2 is odd integer means b = 2 X odd means even

b= {0,1,..9}

Together, b can be {0,2,4,6,8}. We can clearly see yes and no. Insuff
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Re: Number Property - DS Question  [#permalink]

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29 Jul 2009, 19:22
1
Ans is E.

To answe this question, we need to remember that, for any integer to be divisible by 3, the sum of the digits has to be 3 or multiple of 3.

now the given expression = (10^n) + b
whatever be the value of n, the sum of the digits of (10^n) is always 1.
so, the value of b will determine whether (10^n) is divisible by 3.

statement 1 says: b/2 is odd.
which means b is even (even*odd=even). but this is not sufficient.
for example, if b=2, then [(10^n) +2] will have the sum of its digits equal to 3 and hence will be divisible by 3.
but, if b=4, then the sum of digits of [(10^n) +4] equals 5 which is not divisible by 3.
hence statement 1 is not sufficient.

statement 2 says, remainder of b/10 = b, which means b is less than 10.
again, by the above examples we can say this statement also is not sufficient.

combining the two statements we get b is positive even integer less than 10. this also is not sufficient by the same examples.

hence E.
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Re: If a and b are positive integers  [#permalink]

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28 Aug 2015, 11:16
2
sunita123 wrote:
If a and b are positive integers, is (10^a) + b divisible by 3?

1. b/2 is an odd integer.
2. the remainder of b/10 is b

Here is my explanation on this,

Problem statement, 10^a + b is this divisible by 3. In other words, the question is whether the sum of digits that 10^a + b add up to 3. Here whatever value 'a' takes 10^a will only be 1, especially since 'a' is always a positive integer. So this mean, once you add 1 with b's digits, you can solve the question .

Option A,

b/2 is odd, so this means b is 2*(y) where y could be any odd number - 1, 3, 5, 7, 9, etc.... so different values of b are 2, 6, 10, 14, 18... etc... now, find out the sum of digits of 10^a + b ... we know a is always 1, so when b is 1, the sum of digits is 3, when b is 6, value is 7, when b is 10, value is 11, when b is 14, value is 15... so here we notice 6 and 15 are divisible by 3 and rest are not. So this means A is insufficient. <INSUFFICIENT>

Option B,

Remainder of b/10 is b, this means the only possibility is b is less than 10, (check when b is 11, it gives remainder of 1, so the value should only be 9 and less).. so the different values of b are 9, 8, 7, 6, 5.. to 1.... now finding value of sum of digits of 10^a + b, where a is always 1.. so when b is 1, sum is 2, b is 2, sum is 3, when b is 3, sum is 4, when b is 4, sum is 5... etc... If we notice we have values divisible by 3 and those that are not... so this option is insufficient <INSUFFICIENT>

At this point option A, B and D are ruled out

Combine option A and B, this is simple, simply check common values of b for option A and B .. so possible values of b are 2 and 6 only... this makes things easier, we have just calculate sum of digits of 10^a+b for these 2 values.. we had already done this calculation for A and the two sums of digits are 3 and 7 respectively. 3 is divisible by 3 but 7 is not, hence combining also hasn't given us an answer. <INSUFFICIENT>

Hence option is E...

--- Kudos if this explanation helped ------
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Re: If a and b are positive integers, is 10^a + b divisible by 3  [#permalink]

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09 Mar 2019, 06:12
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Re: If a and b are positive integers, is (10^a) + b divisible by  [#permalink]

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21 May 2019, 20:13
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).

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Re: If a and b are positive integers, is (10^a) + b divisible by   [#permalink] 21 May 2019, 20:13
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