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

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
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What is the remainder when 7^30 is divided by 100?  [#permalink]

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06 Dec 2017, 01:12
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Difficulty:

15% (low)

Question Stats:

82% (01:23) correct 18% (02:19) wrong based on 81 sessions

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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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"Only $99 for 3 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" Intern Joined: 06 Oct 2017 Posts: 10 Re: What is the remainder when 7^30 is divided by 100? [#permalink] ### Show Tags 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 D Sent from my vivo 1609 using GMAT Club Forum mobile app Intern Joined: 26 Oct 2017 Posts: 27 Re: What is the remainder when 7^30 is divided by 100? [#permalink] ### Show Tags 06 Dec 2017, 10:56 1 1 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. $$59$$ Attachment: IMG_20171206_232527.jpg [ 32.07 KiB | Viewed 905 times ] Imo D Sent from my ONEPLUS A3003 using GMAT Club Forum mobile app Intern Joined: 11 Nov 2017 Posts: 12 Re: What is the remainder when 7^30 is divided by 100? [#permalink] ### Show Tags 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. _________________ Cheers. Never gonna give up!!!! Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 6248 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: What is the remainder when 7^30 is divided by 100? [#permalink] ### Show Tags 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 _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$99 for 3 month Online Course"
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Joined: 07 Jul 2018
Posts: 7
What is the remainder when 7^30 is divided by 100?  [#permalink]

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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...

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

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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...

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

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