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Bunuel
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M12-27 [#permalink] New post  15 Sep 2014, 23:47
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If \(x10x\) is a four-digit integer, what is digit \(x\), if \(x10x\) is divisible by 36?

A. 2
B. 3
C. 4
D. 6
E. 7
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Bunuel
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Re M12-27 [#permalink] New post  15 Sep 2014, 23:47
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Official Solution:

If \(x10x\) is a four-digit integer, what is digit \(x\), if \(x10x\) is divisible by 36?

A. 2
B. 3
C. 4
D. 6
E. 7

To be divisible by 36, \(x10x\) has to be even and a multiple of 3. This leaves us with \(x = 4\). \(4104 = 36*114\).

Answer: C
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Re: M12-27 [#permalink] New post  02 Sep 2016, 05:47
Bunuel wrote:
Official Solution:

If \(x10x\) is a four-digit integer, what is digit \(x\), if \(x10x\) is divisible by 36?

A. 2
B. 3
C. 4
D. 6
E. 7

To be divisible by 36, \(x10x\) has to be even and a multiple of 3. This leaves us with \(x = 4\). \(4104 = 36*114\).

Answer: C


Great explanation. I used the same apprach, but instead of dividing, I just added the digits (from 1 to 9) to see if the sum is divisible by 3. Obviously, some can be easily discarded like 1 and 5.
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daisy591968
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Re: M12-27 [#permalink] New post  13 Sep 2016, 23:55
2
Bunuel wrote:
Official Solution:

If \(x10x\) is a four-digit integer, what is digit \(x\), if \(x10x\) is divisible by 36?

A. 2
B. 3
C. 4
D. 6
E. 7

To be divisible by 36, \(x10x\) has to be even and a multiple of 3. This leaves us with \(x = 4\). \(4104 = 36*114\).

Answer: C


x10x is divisible by 36, meaning that x10x is divisible by 4 also. The sign of divisible by 4 is that 0x is divisible by 4. So x will be only 4 or 8. Look up all choices, only 4 appears.

We don't need to do any calculation but still find the correct answer.
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Re: M12-27 [#permalink] New post  09 Nov 2016, 14:57
Perhaps, easier number to work with is 9 because 36 is divisible by 9 by looking at the sum of the numbers (3+6)

Thus the sum of digits in the answer choices must be divisible by 9.

Don't even have to write down anything, but doing simple addition in mind conclude us with C. 4+1+0+4 = 9
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TheMastermind
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Re: M12-27 [#permalink] New post  27 Apr 2017, 11:54
Can someone explain why does it have to be divisible by 3 if its divisible by 36?
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Bunuel
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Re: M12-27 [#permalink] New post  28 Apr 2017, 01:56
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TheMastermind wrote:
Can someone explain why does it have to be divisible by 3 if its divisible by 36?


Because 36 = 3*12. If integer x is divisible by integer y, then x is divisible by all factors of y too.
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TheMastermind
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Re: M12-27 [#permalink] New post  28 Apr 2017, 02:15
Bunuel wrote:
TheMastermind wrote:
Can someone explain why does it have to be divisible by 3 if its divisible by 36?


Because 36 = 3*12. If integer x is divisible by integer y, then x is divisible by all factors of y too.


Thank you so much, Bunuel..!
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Re: M12-27 [#permalink] New post  23 Aug 2017, 04:14
x10x is divisible by 36 (2^2*3^2) when x is even and x+1+0+x is divisible by 3.

4 is the answer.
4 is even and 4+1+0+4= 9 which is divisible by 3.
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saurabh1992
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Re: M12-27 [#permalink] New post  05 Jun 2019, 20:36
Hi Bunuel,

Can we also use the below mentioned approach:

To be divisible by 36 "x10x" must be divisible by 4 . So if the last 2 digit is divisible by 4 (i.e 0& x) then its is divisible by 36.

Thanks
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Bunuel
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Re: M12-27 [#permalink] New post  05 Jun 2019, 21:03
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saurabh1992 wrote:
Hi Bunuel,

Can we also use the below mentioned approach:

To be divisible by 36 "x10x" must be divisible by 4 . So if the last 2 digit is divisible by 4 (i.e 0& x) then its is divisible by 36.

Thanks


x10x will be divisible by 4 if x is 0, 4, or 8. Since only 4 is in answer choices, then it must be correct.
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