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

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Senior Manager
Joined: 15 Sep 2011
Posts: 302
Location: United States
WE: Corporate Finance (Manufacturing)
Re: 10^25 – 560 is divisible by all of the following EXCEPT:  [#permalink]

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27 Jun 2015, 10:39
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.
Manager
Joined: 03 Jul 2016
Posts: 65
Re: 10^25 – 560 is divisible by all of the following EXCEPT:  [#permalink]

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04 Aug 2016, 21:04
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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Joined: 07 Jun 2016
Posts: 30
GPA: 3.8
WE: Supply Chain Management (Manufacturing)
Re: 10^25 – 560 is divisible by all of the following EXCEPT:  [#permalink]

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06 Oct 2016, 17:36
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 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.

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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Joined: 03 Sep 2018
Posts: 259
Location: Netherlands
GPA: 4
Re: 10^25 – 560 is divisible by all of the following EXCEPT:  [#permalink]

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14 Jan 2019, 03:25
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]

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09 Feb 2020, 23:40
value of 10^25 has 25 zeroes & 1 one ; last three digits would be 440
440 factors 11*4*5*2 ;
not possible value ; 3
IMO E

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 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?
Intern
Joined: 21 Feb 2018
Posts: 9
Re: 10^25 – 560 is divisible by all of the following EXCEPT:  [#permalink]

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27 Mar 2020, 00:59
This was my approach.

10^25 will be a huge number with 100..000 we know if we subtract the 560 the number will become 9999....460. We have to focus on the 460 and its divisors.

I took 460 and performed a prime factorization, this results in the factors of 11, 5, 2, 2 , 2 (the building blocks to make 460).

I then took to prime factors of the answer choices.

A) 11 - prime factor is 11 and 11 is a prime factor for 460, therefore out.
B) 8 - 2, 2, 2 - 460 has three prime factors of 2, therefore out
C) 5 - this is also a prime factor of 460, therefore out
D) 4 - 2, 2 - 460 contains at least two prime factors of 2 therefore out
E) 3 - is not a prime factor of 460, therefore, cannot be a divisor

Please let me know if there are any errors using this approach.
Re: 10^25 – 560 is divisible by all of the following EXCEPT:   [#permalink] 27 Mar 2020, 00:59

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