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# What is the remainder when 7^100 is divided by 50?

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Math Revolution GMAT Instructor
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What is the remainder when 7^100 is divided by 50?  [#permalink]

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24 May 2018, 18:29
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72% (01:07) correct 28% (01:40) wrong based on 111 sessions

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[GMAT math practice question]

What is the remainder when $$7^{100}$$ is divided by $$50$$?

$$A. 0$$
$$B. 1$$
$$C. 7$$
$$D. 21$$
$$E. 49$$

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MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
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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" Senior Manager Joined: 24 Aug 2016 Posts: 327 Location: India Concentration: Entrepreneurship, Operations GMAT 1: 540 Q49 V16 GMAT 2: 640 Q47 V31 GMAT 3: 630 Q48 V28 GPA: 3.4 Re: What is the remainder when 7^100 is divided by 50? [#permalink] ### Show Tags 24 May 2018, 19:03 MathRevolution wrote: [GMAT math practice question] What is the remainder when $$7^{100}$$ is divided by $$50$$? $$A. 0$$ $$B. 1$$ $$C. 7$$ $$D. 21$$ $$E. 49$$ This question essentially asking what are the last 2 digits of the expression $$7^{100}$$... as what ever is in the hunderds digit, if the last two are 00, the number is always divided by 50. Now cyclicity of 7 is 4, And the numbers are 7,9,3,1........ Hence the last digit is 1 as 25*4=100 now $$7^{4}$$ = 7*7*7*7= 2401 And the expression is $$2401^{25}$$= in that case the last two will be always 01 ( can be tested quickly with $$101^{2}$$ & $$101^{3}$$) Hence the reminder is 01.....................Hence , I would go for option B. _________________ Please let me know if I am going in wrong direction. Thanks in appreciation. Manager Joined: 22 Jun 2017 Posts: 72 Location: Brazil GMAT 1: 600 Q48 V25 GPA: 3.5 WE: Engineering (Energy and Utilities) Re: What is the remainder when 7^100 is divided by 50? [#permalink] ### Show Tags 24 May 2018, 19:04 1 Option B $$7^1$$ = 7 $$7^2$$ = 49 $$7^3$$ = 343 $$7^4$$ = 2401 $$7^5$$ = 16807 $$7^6$$ = 117649 $$7^7$$ = 823543 $$7^8$$ = 5764801 ... So, 7^100 is something that ends with 01. The remander is 1. Manager Joined: 20 Feb 2017 Posts: 166 Location: India Concentration: Operations, Strategy WE: Engineering (Other) Re: What is the remainder when 7^100 is divided by 50? [#permalink] ### Show Tags 24 May 2018, 20:35 2 1 These type of questions become really simple if you understand the concept of negative remainders. Always try and reduce the dividend to 1 or -1. = Rem [7^100 / 50] = Rem [49^50/50] = Rem [ (-1)^50 / 50] = Rem [1 / 50] = 1 hence B _________________ If you feel the post helped you then do send me the kudos (damn theya re more valuable than$)

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Re: What is the remainder when 7^100 is divided by 50?  [#permalink]

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27 May 2018, 18:21
1
1
=>

The remainder when $$7^{100}$$ is divided by $$50$$ depends only on the units and tens digits.

The units digits of $$7^n$$ cycle through the four values $$7, 9, 3$$, and $$1$$.
The tens digits of $$7^n$$ cycle through the four values $$0, 4, 4$$, and $$0$$.

We have the following sequence of units and tens digits for $$7^n$$:

$$7^1 = 07 ~ 07$$
$$7^2 = 49 ~ 49$$
$$7^3 = 343 ~ 43$$
$$7^4 = 2401 ~ 01$$
$$7^5 = 16807~ 07$$

So, $$7^{100} = (7^4)^{25}$$ has the same units and tens digits as $$7^4$$, that is, $$01$$.
Thus, the remainder when $$7^{100}$$ is divided by $$50$$ is $$1$$.

Therefore, B is the answer.

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Re: What is the remainder when 7^100 is divided by 50?  [#permalink]

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29 May 2018, 09:24
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MathRevolution wrote:
[GMAT math practice question]

What is the remainder when $$7^{100}$$ is divided by $$50$$?

$$A. 0$$
$$B. 1$$
$$C. 7$$
$$D. 21$$
$$E. 49$$

We see that 7^2 = 49, which is 50 - 1. Although 49/50 = 0 R 49, rather than using the remainder of 49, let’s call the remainder “-1”.

Since 7^100 = (7^2)^50 = 49^50, which is equivalent to (-1)^50 when it’s divided by 50, and since (-1)^50 = 1, so when (-1)^50 is divided by 50, the remainder is 1.

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What is the remainder when 7^100 is divided by 50?  [#permalink]

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27 Aug 2018, 06:43
$$7^{100}/50=49^{50}/50=(50-1)^{50}/50$$ - only $$(-1)^{50} = 1^{50}=1$$ - won't be devisable by 50. The remainder is 1.

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Re: What is the remainder when 7^100 is divided by 50?  [#permalink]

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23 Sep 2018, 05:00
ScottTargetTestPrep wrote:
MathRevolution wrote:
[GMAT math practice question]

What is the remainder when $$7^{100}$$ is divided by $$50$$?

$$A. 0$$
$$B. 1$$
$$C. 7$$
$$D. 21$$
$$E. 49$$

We see that 7^2 = 49, which is 50 - 1. Although 49/50 = 0 R 49, rather than using the remainder of 49, let’s call the remainder “-1”.

Since 7^100 = (7^2)^50 = 49^50, which is equivalent to (-1)^50 when it’s divided by 50, and since (-1)^50 = 1, so when (-1)^50 is divided by 50, the remainder is 1.

Hi,

thanks for this solution, but I have a doubt. this question doesn't say that there is exponent for 50. then, How can we take (-1)^50 ?

Regards,
Kishlay
Re: What is the remainder when 7^100 is divided by 50? &nbs [#permalink] 23 Sep 2018, 05:00
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# What is the remainder when 7^100 is divided by 50?

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