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Of the following, which is the largest number that is a factor of 101^

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Bunuel
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Of the following, which is the largest number that is a factor of 101^ [#permalink] New post   06 Oct 2019, 01:14
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A
B
C
D
E

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95% (hard)

Question Stats:

13% (02:00) correct 88% (01:54) wrong based on 200 sessions
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Of the following, which is the largest number that is a factor of \(101^{100} - 1\) ?

(A) 100
(B) 10,000
(C) 100,100
(D) 100,000
(E) 10,000,000


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The largest number amongst the following that will perfectly divide
101^100 - 1 is
­
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B
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nick1816
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Re: Of the following, which is the largest number that is a factor of 101^ [#permalink] New post   06 Oct 2019, 01:44
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\((100+1)^{100}-1\)= 100C0 \(100^{100}\)+ 100C1 \(100^{99}\)+...........+100C99 *100 + 100C100 -1

\((100+1)^{100}-1\)= 100C0 \(100^{100}\)+ 100C1 \(100^{99}\)+...........+100 *100

\((100+1)^{100}-1\)= \(10^4\) ( 100C0 \(100^{96}\)+ 100C1 \(100^{95}\)+...........+1)

Hence Maximum power of 10 that can divide \((100+1)^{100}-1\) is 4

Eliminate A, D and E

100100=7*11*13*100

101=3 mod 7
\(101^{100}\)= \(3^{100}\) mod 7
\(101^{100}\)= \(9^{50}\) mod 7
\(101^{100}\)= \(2^{50}\) mod 7
\(101^{100}\)= \(8^{16}*4\) mod 7
\(101^{100}\)= 4 mod 7
\(101^{100}-1\)= 3 mod 7

Hence \(101^{100}-1\) can't be divisible by 7 or 100100

Eliminate C

B






Bunuel wrote:

TOUGH QUESTION



Of the following, which is the largest number that is a factor of \(101^{100} - 1\) ?

(A) 100
(B) 10,000
(C) 100,100
(D) 100,000
(E) 10,000,000


Show Spoiler
The largest number amongst the following that will perfectly divide
101^100 - 1 is
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adityamegh3009
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Re: Of the following, which is the largest number that is a factor of 101^ [#permalink] New post   26 Dec 2020, 06:04
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I think there is an easy answer to it :- (100+1)^100 -1 when expanded using Binomial theorem the last term is 1 which will get cancelled by 1 and the second last term is 100C99*100. All other terms are >than this. Thus B

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Re: Of the following, which is the largest number that is a factor of 101^ [#permalink] New post   08 Oct 2019, 22:27
Not 100% sure but here goes:
101^2 - 1 = 10200
101^3 - 1 = 1030300.....
101^10 - 1 = .........91000
101^11 - 1 = .........9191100
101^12 - 1 =.........301200

There is a pattern here: The hundred digit of the RHS is equal to the power to which 101 is raised. The hundred digit continues to increase until 101 is raised to the power 10 when the hundred digit becomes 0 and the 1000 digit becomes 1 thus rendering the RHS divisible by 1000. But when 101 is raised to the power 11, the hundred digit of the RHS again becomes 2 (thus making 100 its highest factor) and continues to increase to 9 until 101 is raised to the power 20 at which time the highest factor of the RHS again becomes 1000.
Since this pattern repeats itself, when 101 is raised to hundred and 1 is subtracted, the highest number it will be divisible by is 1000 but since this is not in the answer options, we have to settle for 100. ANS: A
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Re: Of the following, which is the largest number that is a factor of 101^ [#permalink] New post   21 Mar 2024, 11:41
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Re: Of the following, which is the largest number that is a factor of 101^ [#permalink]
21 Mar 2024, 11:41
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