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What is the remainder when 7^30 is divided by 100?

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What is the remainder when 7^30 is divided by 100?  [#permalink]

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New post 06 Dec 2017, 01:12
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A
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D
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Question Stats:

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[GMAT math practice question]

What is the remainder when \(7^{30}\) is divided by \(100\)?

A. \(19\)
B. \(29\)
C. \(39\)
D. \(49\)
E. \(59\)

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Re: What is the remainder when 7^30 is divided by 100?  [#permalink]

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New post 06 Dec 2017, 06:57
2
7 has cyclicity of 4

7^4 =2401, when divided by 100 will leave remainder as 1

((7^4)^7*7^2)/100
=> 1*49/100
Remainder =49

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Re: What is the remainder when 7^30 is divided by 100?  [#permalink]

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New post 06 Dec 2017, 10:56
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MathRevolution wrote:
[GMAT math practice question]

What is the remainder when \(7^{30}\) is divided by \(100\)?

A. \(19\)
B. \(29\)
C. \(39\)
D. \(49\)
E. \(59\)

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Re: What is the remainder when 7^30 is divided by 100?  [#permalink]

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New post 06 Dec 2017, 12:03
MathRevolution wrote:
[GMAT math practice question]

What is the remainder when \(7^{30}\) is divided by \(100\)?

A. \(19\)
B. \(29\)
C. \(39\)
D. \(49\)
E. \(59\)



Straight D.

Remainders when divided by 100 are
7^1 = 7
7^2 = 49
7^3 = 43
7^4 = 1
7^5 = 7

So, cyclicity of 4. 30%4=2
Remainder when divided by 100 is 49.
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Re: What is the remainder when 7^30 is divided by 100?  [#permalink]

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New post 08 Dec 2017, 01:07
1
=>

The tens digit of \(7^k\) cycles through \(0-->4-->4--->0--->0-->…..\)
The units digit of \(7^k\)cycles through \(7-->9-->3--->1--->7-->…..\)

Since \(30=4(7)+2, 7^{30}=7^{4(7)+2}\) ends in \(49\).

Therefore, the answer is D.

Answer: D
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What is the remainder when 7^30 is divided by 100?  [#permalink]

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New post 08 Jul 2018, 15:22
Question for the experts:
How does the cyclicity work?

I wonder if the cyclicity of the "powered up" numbers goes well beyond units or tens digits. Actually once I've computed one power can I reuse it? Illustration [inspired by math revolution post]:

The thousands digits of 7^k cycles through......?−−>?−−>?−−−>2−−−>?−−>…
The hundreds digits of 7^k cycles through.......?−−>?−−>3−−−>4−−−>?−−>…
The tens digit of 7^k cycles through.............0−−>4−−>4−−−>0−−−>0−−>…
The units digit of 7^k cycles through............7−−>9−−>3−−−>1−−−>7−−>…

The only calculation I've made are 7^3 and 7^4 and with that I've partly filled the 7 cycle up to the thousands! But then I've checked for 2401^2 but it doesn't work...

Help please... Thx
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Re: What is the remainder when 7^30 is divided by 100?  [#permalink]

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New post 09 Jul 2018, 07:51
jetmat wrote:
Question for the experts:
How does the cyclicity work?

I wonder if the cyclicity of the "powered up" numbers goes well beyond units or tens digits. Actually once I've computed one power can I reuse it? Illustration [inspired by math revolution post]:

The thousands digits of 7^k cycles through......?−−>?−−>?−−−>2−−−>?−−>…
The hundreds digits of 7^k cycles through.......?−−>?−−>3−−−>4−−−>?−−>…
The tens digit of 7^k cycles through.............0−−>4−−>4−−−>0−−−>0−−>…
The units digit of 7^k cycles through............7−−>9−−>3−−−>1−−−>7−−>…

The only calculation I've made are 7^3 and 7^4 and with that I've partly filled the 7 cycle up to the thousands! But then I've checked for 2401^2 but it doesn't work...

Help please... Thx


Please go through the discussion here

https://gmatclub.com/forum/cyclicity-on ... 13019.html
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Re: What is the remainder when 7^30 is divided by 100? &nbs [#permalink] 09 Jul 2018, 07:51
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