What is the remainder when 25^25 is divisible by 26 ?

Question

(A) 1 
(B) 2
(C) 3
(D) 24 
(E) 25

DHRJ0032 · Accepted Answer

Using negative remainders concept,
25²⁵/26 can be written as 
(-1)²⁵/26
=-1/26
=25

E it is

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Bunuel · Answer

The method used above is the best one.

25^25 = (26 - 1)^25

When you expand this expression, all terms except the last one will have 26 as their multiple and thus will be divisible by 26. The last one will be (-1)^25 = -1. Hence, the question essentially boils down to finding the remainder when -1 is divided by 26. -1 divided by 26 yields the remainder of 25.

Answer: E.

P.S. The process for finding the remainder when dividing a negative integer by a positive integer follows the same principles as when dividing a positive integer by a positive integer.

For example:
 
 What is the remainder when dividing 21 by 6? We find the closest multiple of 6 that is  less  than 21, which is 18. Then, we calculate 21 - 18, yielding a remainder of 3. 
 What is the remainder when dividing -23 by 7? Here, we find the closest multiple of 7 that is  less  than -23, which is -28. Then, we calculate -23 - (-28), resulting in a remainder of 5.

What about dividing -1 by 26? The closest multiple of 26  less  than -1 is -26. So, the remainder is -1 - (-26) = 25.

Alternatively, consider this method:
 
 What is the remainder when dividing -23 by 7? Dividing 23 by 7 gives a remainder of 2. To find the remainder for -23 divided by 7, subtract this 2 from the divisor. Thus, the remainder when dividing -23 by 7 is 7 - 2 = 5.

Similarly, what is the remainder when dividing -1 by 26? Dividing 1 by 26 gives a remainder of 1. Therefore, the remainder when dividing -1 by 13 is 26 - 1 = 25.

Hope it helps.

Makhabbat · Answer

Hello! Please help, is there different solution or maybe someone can explain this solution with more details?

rajshreeasati · Answer

Makhabbat 
we have 25^25 to be divided by 26.
Look at 25 in isolation, without the power => so 25 div. by 26 gives remainder 25.
now, we need remainder such that when remainder of 25/26^25 div. by 26 will be our final remainder

now 25^1 /26, remainder = 25
25^2 /26, remainder = 1
thus we can write {(25^2)^12 * 25}/26 thus final remainder = ((25^2)^11/26) * (25/26)R = 1 * 25 =>  25 . final ans.

Abhishek009 · Answer

\frac{25^25}{26}

=\frac{(26-1)^{25}}{26}

Rem is  1 - 26 = 25  , Answer must be (E)

as19 · Answer

Abhishek009

May you please developed. Bit more your answer?

I tried it your way but got confused because, as a I see it :

(26-1)^25 / 26

26^25 - 1^25 / 26

Remainder when 26^25/26 is 0 
Remainder when 1^25/26 is 1

So my answer was 1 (A)

Where am I getting wrong on my process?

Thank you

Abhishek009 · Answer

Please go through this thread to brush up on the concept of the negative remainder from our legendry admin Bunuel https://gmatclub.com/forum/all-about-ne ... l#p1472077 In case of any issue please feel free to revert, we will be more than happy to dissect the question.

Mansouri · Answer

Hello  Bunuel , ! Please, can you explain this question with more details or alternate method?

Fish181 · Answer

Simplified Explanation:

When the base is just under the modulus, (26) the remainder will always just be the base if its raised to an odd exponent.

If it's raised to an even exponent then it will be 1.

If the exponent is **odd**, like $25^{25}$, the result is $n-1$ when $a = n-1$. So, $25^{25} \mod 26 = 25$.

This "trick" is part of modular arithmetic.

You're basically looking at how much is left over when you try to share something almost the same size as what you're dividing by. Since 25 is just one less than 26, if you multiply it by itself a bunch of times, you're still going to have that "just under" situation.

This is a neat trick you can use for any numbers like this, not just 25 and 26. It’s a fast way to solve these kinds of problems without doing a lot of calculations.

gmattaker16 · Answer

(26-1)^25 is not 26^25 - 1^25. (a-b)^ = a^2 + b^2 - 2ab. You missed the -2ab.

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What is the remainder when 25^25 is divisible by 26 ?

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What is the remainder when 25^25 is divisible by 26 ?

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