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10^25 – 560 is divisible by all of the following EXCEPT:

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Senior Manager
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Re: 10^25 – 560 is divisible by all of the following EXCEPT:  [#permalink]

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New post 27 Jun 2015, 11:39
Here's a trick to calculate whether the number is a mulitple of 11 (and 7 or 13):

1. Seperate the number from right to left in groups of 3 digits
2. Add the groups in the even and odd positions separately.
3. If the difference between the sums is divisible by 7, 11, or 13, then the entire number is divisible by 7, 11, or 13.

Since the all the digits from the 7th digit through the 22nd digit are nine and the difference is zero, the first two groups of three will be sufficient. So, 999 - 440 = 550 and 550/11 = integer, and therefore is divisible by 11.
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Re: 10^25 – 560 is divisible by all of the following EXCEPT:  [#permalink]

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New post 04 Aug 2016, 22:04
Here is my approach :

10^25 - 560 --> 5x2^4(10^21X125 - 7) . We can rule out B, C, D options.

5x2^4(10^21X125 - 7) = 124999..3 is no way divisible by 3 but can be divisible by 11.

So 'E'
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Re: 10^25 – 560 is divisible by all of the following EXCEPT:  [#permalink]

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New post 06 Oct 2016, 18:36
Bunuel wrote:
enigma123 wrote:
10^25 – 560 is divisible by all of the following EXCEPT:
a)11
b)8
c)5
d)4
e) 3

Guys any idea what concept has been Tested over here and what will be the answer?

I have started doing it this way but got stuck. So can someone please help?

I have started from

10^5 - 560 = 99,440 i.e. it has two 9s followed by 440.
.
.
10^10 - 560 = 99,99,440 ------------------------------> Am I doing it right this way?


Yes, you were on a right track.

10^(25) is a 26 digit number: 1 with 25 zeros. 10^(25)-560 will be 25 digit number: 22 9's and 440 in the end: 9,999,999,999,999,999,999,999,440 (you don't really need to write down the number to get the final answer). From this point you can spot that all 9's add up to some multiple of 3 (naturally) and 440 add up to 8 which is not a multiple of 3. So, the sum of all the digits is not divisible by 3 which means that the number itself is not divisible by 3.

Answer: E.

You can also quickly spot that the given number is definitely divisible:
By 2 as the last digit is even;
By 4 as the last two digits are divisible by 4;
By 8 as the last three digits are divisible by 8;
By 11 as 11 99's as well as 440 have no reminder upon division by 11 (or by applying divisibility by 11 rule).

Check Divisibility Rules chapter of Number Theory: math-number-theory-88376.html


Thank you Bunuel. I used this tactic after seeing you reference it in other posts. I learned this before but had not committed it to memory until now. Great explanation as always
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Re: 10^25 – 560 is divisible by all of the following EXCEPT:  [#permalink]

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New post 14 Jan 2019, 04:25
I am wondering whether the following test for divisibility by 11 is valid:

\((11-1)^{25}-560\) \(\implies\) \(Remainders: -1-10=-11\) \(\implies\) \(Remainder_{total} = 0\)

I am unsure whether from \((11-1)^{25}\) it follows that the remainder is \(-1\) as for instance the remainder of \((11-1)^2\) is \(-10\). So how do I know that the remainder is only -1 when the exponent of \((11-1)^{25}\) is odd?
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Re: 10^25 – 560 is divisible by all of the following EXCEPT:   [#permalink] 14 Jan 2019, 04:25

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