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

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

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

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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.
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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
2
1
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
1
$$7^{100}/50=49^{50}/50=(50-1)^{50}/50$$ - 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?   [#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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