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

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mejia401
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Re: 10^25 – 560 is divisible by all of the following EXCEPT: [#permalink] New post  27 Jun 2015, 11:39
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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] 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] 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] 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] New post  10 Feb 2020, 00:40
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


Show Spoiler
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?
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Re: 10^25 – 560 is divisible by all of the following EXCEPT: [#permalink] New post  27 Mar 2020, 01:59
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.
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Re: 10^25 – 560 is divisible by all of the following EXCEPT: [#permalink] New post  07 Apr 2020, 00:23
Fast Approach:
10^25 – 560 is divisible by all of the following EXCEPT:

A. 11
B. 8
C. 5
D. 4
E. 3


We know that 10^25 finish with zero.

Suppose 1000 - 560 = 440.

440 is an even number, so we can eliminate all the even answer choices: B and D.
440 is also a multiple of 5, so we can eliminate C.

We remain with A (11) and E (3).
440 can be expressed as: 11*40, thus 440 is divisible by 11.

We remain with only 3.

Answer E.
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Re: 10^25 – 560 is divisible by all of the following EXCEPT: [#permalink] New post  07 Apr 2020, 01:16
10^25 = 3a +1
Where a is the natural numbers
560 =3b +2
Where b is the natural number
10^25- 560
3a +1- 3b - 2
3(a-b) -1
Not divisible by 3
Option E is the answer

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Re: 10^25 – 560 is divisible by all of the following EXCEPT: [#permalink] New post  15 Sep 2020, 05:21
10^25 – 560
will give us 22*9 & 440 in end
given is even so options b,c,d can be eliminated
among a & e ; 22 times 9 and 440 would be divisble by 11
left with 3 which does not divide the value
option E


enigma123 wrote:
10^25 – 560 is divisible by all of the following EXCEPT:

A. 11
B. 8
C. 5
D. 4
E. 3


Show Spoiler
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?
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Re: 10^25 – 560 is divisible by all of the following EXCEPT: [#permalink] New post  26 Oct 2020, 14:51
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10^25 – 560 is divisible by all of the following EXCEPT:

A. 11
B. 8
C. 5
D. 4
E. 3

10^25 - 560 = 999999...999440

Right off the bat we can eliminate B, C, D,

The question is between 11 and 3.

Is the number divisible by 3? The divisibility rule of 3 is that if the sum of the digits is divisible by 3, the number is also divisible by 3.

Multiple of 9 + 4 + 4 = not divisible by 3.

Answer is E.
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Nipungupta9081
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Re: 10^25 – 560 is divisible by all of the following EXCEPT: [#permalink] New post  29 Oct 2020, 13:08
enigma123 wrote:
10^25 – 560 is divisible by all of the following EXCEPT:

A. 11
B. 8
C. 5
D. 4
E. 3



My Approach is Kinda Different.

Rather then multiplying 10^25 we can just take 1000 - 560 which will yield = 440

Which means it is divisible by all numbers except 3

Answer = E
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Re: 10^25 560 is divisible by all of the following EXCEPT: [#permalink] New post  20 May 2023, 15:17
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: https://gmatclub.com/forum/math-number- ... 88376.html


Hi

Why does the bit above in bold show that the number has no remainder upon division by 11?
Thanks
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Re: 10^25 560 is divisible by all of the following EXCEPT: [#permalink]
20 May 2023, 15:17
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