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study
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If a and b are positive integers, is 10^a + b divisible by 3 Updated on: 12 Oct 2013, 09:15
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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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.

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

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If a and b are positive integers, is 10^a + b divisible by 3 29 Jul 2009, 19:22
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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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If a and b are positive integers, is 10^a + b divisible by 3 25 May 2023, 13:04
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Official Solution:


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

Note that \(10^a\), where \(a\) is a positive integer, will always be 1 more than a multiple of 3: 10, 100, 1,000, and so on. Therefore, for \(10^a + b\) to be divisible by 3, \(b\) must be one less than a multiple of 3, i.e., 2, 5, 8, 11, 14, and so on. In this case, \(10^a + b = (\text{multiple of 3} + 1) + (\text{multiple of 3} - 1) = 2*\text{multiple of 3}\), making \(10^a + b\) divisible by 3.

(1) \(\frac{b}{2}\) is an odd integer.

The above statement implies that \(b=2* \text{odd integer}\). Therefore, \(b\) could be 2, 6, 10, 14, 18, and so on. Thus, \(b\) could be one less than a multiple of 3, for example 2 or 14. However, it could also be a multiple of 3, such as 6 or 18, as well as two less than a multiple of 3, such as 10 or 22. Not sufficient.

(2) The remainder of \(\frac{b}{10}\) is \(b\).

This statement implies that \(b\) is less than 10: i.e., 1, 2, 3, ..., or 9. However, this information is also insufficient to determine if \(10^a + b\) is divisible by 3.

(1)+(2) Statement (1) tells us that \(b\) is of the form \(2* \text{odd integer}\), while statement (2) tells us that \(b\) is less than 10. As a result, \(b\) could be 2 or 6. If \(b = 2\) (i.e., one less than a multiple of 3), then \(10^a + b\) will be divisible by 3. However, if \(b = 6\) (a multiple of 3), then \(10^a + b\) will not be divisible by 3. Therefore, even together, the statements are insufficient to determine whether \(10^a + b\) is divisible by 3.


Answer: E
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If a and b are positive integers, is 10^a + b divisible by 3 23 Nov 2024, 07:32
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