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What is the remainder when 25^25 is divided by 9?

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Bunuel
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What is the remainder when 25^25 is divided by 9? [#permalink] New post   21 Feb 2020, 06:13
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What is the remainder when \(25^{25}\) is divided by 9?

A. 7
B. 5
C. 3
D. 2
E. 1


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What is the remainder when 25^25 is divided by 9? [#permalink] New post   31 May 2020, 23:50
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minirana wrote:
lacktutor wrote:
What is the remainder when \(25^{25}\) is divided by 9?

\(25^{25}= (18+7)^{25}\) --> \(7^{25}= 7*7^{24}= 7( 49^{12})= 7*(45+4)^{12})\)

--> \(7*4^{12}= 7(63+1)^{4} = 7(....+1)^{4}\)

The remainder will be 7

The answer is A.


Hi,

My doubt is what thought process went into you deciding to break up 25 into 18+7 and not 24+1 ?

The reasons is as mentioned below:




Thanks,
Mini.


minirana

Please check the video and understand the reason why 25 was written 18+7 instead of 24+1

If you like the video and want to watch many such videos then please subscribe to my YouTube channel :)

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What is the remainder when 25^25 is divided by 9? [#permalink] New post   Updated on: 23 Feb 2020, 22:18
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What is the remainder when \(25^{25}\) is divided by 9?

\(25^{25}= (18+7)^{25}\) --> \(7^{25}= 7*7^{24}= 7( 49^{12})= 7*(45+4)^{12})\)

--> \(7*4^{12}= 7(63+1)^{4} = 7(....+1)^{4}\)

The remainder will be 7

The answer is A.

Originally posted by lacktutor on 21 Feb 2020, 07:00.
Last edited by lacktutor on 23 Feb 2020, 22:18, edited 1 time in total.
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What is the remainder when 25^25 is divided by 9? [#permalink] New post   21 Feb 2020, 07:16
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25^25 = 5^50
Now, 5/9 = remainder 5
5^2/9 = 25/9 = remainder 7
5^3/9 = 125/9 = remainder 8 or remainder (-1)

5^50 = 5^48 * 5^2 = (5^3)^16 * 5^2

Remainder of 5^50 /9 = remainder ((5^3)^16/9) * remainder (5^2/9)
= (-1)^16 * 7 = 7, Option A
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Re: What is the remainder when 25^25 is divided by 9? [#permalink] New post   21 Feb 2020, 07:20
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Bunuel wrote:
What is the remainder when \(25^{25}\) is divided by 9?

A. 7
B. 5
C. 3
D. 2
E. 1
Are You Up For the Challenge: 700 Level Questions


25/7 - remainder 7 (remainder = -2)
25^2/7 - remainder (4)
25^3/7 - remainder = -2^3 = -8 = -8+7 = -1

Therfore 25^24 = (-1)^8 = 1 remainder

Therefore 25^25 = 1*7 = 7 remainder

Answer - A
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Re: What is the remainder when 25^25 is divided by 9? [#permalink] New post   23 Feb 2020, 05:40
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Bunuel wrote:
What is the remainder when \(25^{25}\) is divided by 9?

A. 7
B. 5
C. 3
D. 2
E. 1


Are You Up For the Challenge: 700 Level Questions


\(25^{25} = 5^{50} = 5^2*5^{48} = 25*(5^3)^{16} = 25(126 - 1)^{16}\)
\(25(126 - 1)^{16} = 25(9k + 1) = 9m + 25\), for some positive integers \(k\) & \(m\)

--> \(25^{25} = 9m + 25 = 9(m + 2) + 7\)
--> Remainder when \(25^{25}\) divided by \(9\) = \(7\)

Option A
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Re: What is the remainder when 25^25 is divided by 9? [#permalink] New post   06 May 2020, 22:30
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Hi
25 ^(Anything>0) gives us the last two digits of the number as 25.
Hence remainder is 25/9 --> 7

Hope this helps!
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What is the remainder when 25^25 is divided by 9? [#permalink] New post   06 May 2020, 23:37
GMAT0010 wrote:
Hi
25 ^(Anything>0) gives us the last two digits of the number as 25.
Hence remainder is 25/9 --> 7

Hope this helps!


hi, GMAT0010
\(25^{1}= 25 = 9*2\)+ 7
\(25^{2}= 625= 69*9\) +4

You think that the remainders are the same ???
Hope it helps
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Re: What is the remainder when 25^25 is divided by 9? [#permalink] New post   07 May 2020, 05:19
lacktutor

Understood. Thanks for correcting me.
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Re: What is the remainder when 25^25 is divided by 9? [#permalink] New post   31 May 2020, 23:27
lacktutor wrote:
What is the remainder when \(25^{25}\) is divided by 9?

\(25^{25}= (18+7)^{25}\) --> \(7^{25}= 7*7^{24}= 7( 49^{12})= 7*(45+4)^{12})\)

--> \(7*4^{12}= 7(63+1)^{4} = 7(....+1)^{4}\)

The remainder will be 7

The answer is A.


Hi,

My doubt is what thought process went into you deciding to break up 25 into 18+7 and not 24+1 ?

Thanks,
Mini.
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Re: What is the remainder when 25^25 is divided by 9? [#permalink] New post   15 Aug 2020, 09:43
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Asked: What is the remainder when \(25^{25}\) is divided by 9?
25 = 27 - 2
25mod9 = - 2

\(25^{25}mod9 = (-2)^{25}mod9 = -2^{3*8+1}mod9 = -(-1)^8*2mod9 = -2mod9 = 7mod9\)

IMO A
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Re: What is the remainder when 25^25 is divided by 9? [#permalink] New post   05 May 2021, 05:25
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Hello, math experts.Can you please help me figure out where I am going wrong?

25^25=(27-2)^25

Now,(27-2)^25/9=2^25/9=(2^24*2)/9={(9-1)^8*2}/9=2/9
remainder=2
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Re: What is the remainder when 25^25 is divided by 9? [#permalink] New post   05 May 2021, 05:27
Is there any specific method to attempt these questions?
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Re: What is the remainder when 25^25 is divided by 9? [#permalink] New post   06 May 2021, 11:30
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Jihyo wrote:
Hello, math experts.Can you please help me figure out where I am going wrong?

25^25=(27-2)^25

Now,(27-2)^25/9=2^25/9=(2^24*2)/9={(9-1)^8*2}/9=2/9
remainder=2

I am such a fool... I wrote 2^25 instead of -2^25 .. I am quoting this so that no one does such a horrible mistake while taking the real exam.
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Re: What is the remainder when 25^25 is divided by 9? [#permalink] New post   03 Nov 2021, 14:14
Bunuel wrote:
What is the remainder when \(25^{25}\) is divided by 9?

A. 7
B. 5
C. 3
D. 2
E. 1


Are You Up For the Challenge: 700 Level Questions



\(= \frac{(27-2)^{25}}{9}\)

\(= \frac{(-2)^{25}}{9}\)

Find power of 2 such that it is of the form denominator+1 or denominator-1

\(= \frac{(-2)^{24}*-2}{9}\)

\(= \frac{(-8)^{8}*-2}{9}\)

\(= \frac{(-1)^{8}*-2}{9}\)

\(= \frac{-2}{9}\)

= 7
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Re: What is the remainder when 25^25 is divided by 9? [#permalink] New post   11 Nov 2021, 21:26
GMAT0010 wrote:
Hi
25 ^(Anything>0) gives us the last two digits of the number as 25.
Hence remainder is 25/9 --> 7

Hope this helps!


what about 25 and 125? Your method is easily proven wrong
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Re: What is the remainder when 25^25 is divided by 9? [#permalink] New post   20 Nov 2023, 10:22
\((-2)^{25}=(2)^{24}(-2)=(-2)(8)^8=(-2)(9-1)^8=(-2)(1)=-2+9=7\)
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Re: What is the remainder when 25^25 is divided by 9? [#permalink]
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