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# a and b are positive integers less than or equal to 9. If a

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a and b are positive integers less than or equal to 9. If a [#permalink]

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03 Jan 2013, 05:47
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a and b are positive integers less than or equal to 9. If a and b are assembled into the six-digit number ababab, which of the following must be a factor of ababab?

(A) 3
(B) 4
(C) 5
(D) 6
(E) none of the above
[Reveal] Spoiler: OA

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Last edited by Bunuel on 03 Jan 2013, 06:14, edited 1 time in total.
Edited the question.
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Re: a and b are positive integers less than or equal to 9. If a [#permalink]

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03 Jan 2013, 06:20
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daviesj wrote:
a and b are positive integers less than or equal to 9. If a and b are assembled into the six-digit number ababab, which of the following must be a factor of ababab?

(A) 3
(B) 4
(C) 5
(D) 6
(E) none of the above

The sum of the digits of the six-digit number ababab is a+b+a+b+a+b=3(a+b), thus the sum of the digits is divisible by 3, which means that ababab must be divisible by 3.

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Re: a and b are positive integers less than or equal to 9. If a [#permalink]

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03 Jan 2013, 10:02
Since ab is in the number ababab 3 times and if the sum of any number is a multiple of 3 then 3 must be a factor of that number, the only answer choice that MUST be a factor of ababab is 3. Answer - A
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Re: a and b are positive integers less than or equal to 9. If a [#permalink]

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25 Aug 2014, 04:02
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Re: a and b are positive integers less than or equal to 9. If a [#permalink]

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26 Jun 2015, 20:25
Dear Bunnel,

What about if I take a=4 b=5.

It's not divisible by 3 please assist.
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Re: a and b are positive integers less than or equal to 9. If a [#permalink]

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26 Jun 2015, 21:52
It works even when you use 4 and 5.
4=a
5=b
so the number is 454545.
Adding up all 3(4) + 3(5) = 3(9) = 27.
This is divisible by 3.
The of the question essence lies the data point that ab is repeated thrice.
hence should be divisible by 3
Hope this helps
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a and b are positive integers less than or equal to 9. If a [#permalink]

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26 Jun 2015, 23:59
vikasbansal227 wrote:
Dear Bunnel,

What about if I take a=4 b=5.

It's not divisible by 3 please assist.

454545 when divided by 3 gives the result 151515

$$\frac{454545}{3} = 151515$$

CONCEPT: Rule of Divisibility of 3 is "If sum of the digits of the Number is divisible by 3 then the number will be divisible by 2

SUm of the digits of No. 454545 is 27 which is divisible by 3 hence the Number 454545 will be divisible by 3

I hope it helps!
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Re: a and b are positive integers less than or equal to 9. If a [#permalink]

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02 Jul 2016, 10:49
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Re: a and b are positive integers less than or equal to 9. If a [#permalink]

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13 Mar 2017, 02:55
(A) 3: ababab is divisible by 3 because the sum of its digits is 3(a + b), a multiple of 3 for any
integers a and b.
(B) 4: An integer is divisible by 4 if its last two digits represent a two-digit number that is itself
divisible by 4. It is uncertain whether the two-digit integer ab is divisible by 4.
(C) 5: An integer is divisible by 5 if the last digit is 0 or 5. It is uncertain whether the positive
integer b is 5.
(D) 6: An integer is divisible by 6 if it is even and divisible by 3. We already established that ababab is divisible by 3, but it is uncertain whether the last digit b is even, a requirement for ababab to be even.

Alternatively, we can tackle this problem by thinking about the place values of the unknowns. If we had a two-digit number ab, we could express it as 10a + 1b. By similar logic, ababab can be expressed as follows:
ababab = 100,000a + 10,000b + 1,000a + 100b + 10a + b
If we combine like terms, we get the following:
ababab = 101,010a + 10,101b

At this point, we can spot a common term: each term is a multiple of 10,101. If we factor 10,101 from each term, the expression can be written as follows:
ababab = 10,101(10a + b), where a and b are individual digits.

Or simply:

ababab = 10,101(ab), where ab is a two-digit number.
Since we don't know the value of the two-digit number ab, we cannot know what its factors are. To find a known factor of ababab, our only option is to find a factor of 10,101.
At this point, we can recognize that 10,101 is a multiple of 3 (the sum of the digits is 3). Therefore, ababab must be a multiple of 3.

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Re: a and b are positive integers less than or equal to 9. If a   [#permalink] 13 Mar 2017, 02:55
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