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

Question

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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GMATinsight · Accepted Answer

minirana

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

https://www.youtube.com/watch?v=AZwrtCIcgo0

lacktutor · Answer

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.

Rinng0 · Answer

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

shameekv1989 · Answer

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

CareerGeek · Answer

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

GMAT0010 · Answer

Hi
25 ^(Anything>0) gives us the last two digits of the number as 25.
Hence remainder is 25/9 --> 7

Hope this helps!

lacktutor · Answer

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

GMAT0010 · Answer

lacktutor

Understood. Thanks for correcting me.

minirana · Answer

Hi,

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

Thanks,
Mini.

Kinshook · Answer

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

Jihyo · Answer

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

Jihyo · Answer

Is there any specific method to attempt these questions?

Jihyo · Answer

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.

danbrown9445 · Answer

= \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

langyuwang · Answer

what about 25 and 125? Your method is easily proven wrong

Oppenheimer1945 · Answer

(-2)^{25}=(2)^{24}(-2)=(-2)(8)^8=(-2)(9-1)^8=(-2)(1)=-2+9=7

BrushMyQuant · Answer

We need to find the remainder when  25^{25}  is divided by 9? 
 
 We solve these problems by using Binomial Theorem, where we split the number into two parts, one part is a multiple of the divisor(9) and a big number, other part is a small number.

=>  25^{25}  =  (27-2)^{25}

Watch this video to  MASTER BINOMIAL Theorem

Now, if we use Binomial Theorem to expand this then all the terms except the last term will be a multiple of 9
=> All terms except the last term will give remainder of 0 when divided by 9
=> Problem is reduced to what is the remainder when the last term (i.e. 25C25 * 27^0 * (-2)^25) is divided by 9
=> Remainder of -2^25 is divided by 9

To solve this problem we need to find the cycle of remainder of power of 2 when divided by 9

Remainder of  2^1  (=2) by 9 = 2
Remainder of  2^2  (=4) by 9 = 4 
Remainder of  2^3  (=8) by 9 = 8
Remainder of  2^4  (=16) by 9 = 7
Remainder of  2^5  (=32) by 9 = 5
Remainder of  2^6  (=64) by 9 = 1
 Remainder of  2^7  (=64) by 9 = 2

=> Cycle is 6

=> We need to find Remainder of 25 by 6 = 1
=> Remainder of  2^{25}  is divided by 9 = Remainder of 2^1 by 9 = 2
   => Remainder of  -2^{25}  is divided by 9 = -2 = -2 + 9 = 7

So,  Answer will be A 
Hope it Helps!

Watch following video to MASTER Remainders by 2, 3, 5, 9, 10 and Binomial Theorem

https://www.youtube.com/watch?v=h1HTEhmUb1w

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

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

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