What is the remainder when 3^32 is divided by 82?

Question

What is the remainder when  3^{32}  is divided by 82?

A) 0 
B) 1
C) 8
D) 9
E ) 81

Kinshook · Accepted Answer

Asked: What is the remainder when  3^{32}  is divided by 82?

Remainder when 3^4 = 81 is divided by 82 = - 1
Remainder when (3^4)^8 is divided by 82 = (-1)^8 = 1

IMO B

shameekv1989 · Answer

3^{32} = (3^{4})^{8}

3^4  = 81

\frac{81}{82} => remainder = 81 => remainder = -1

Therefore remainder =  -1^8  = 1 (Property :- Remainders multiply)

Answer - B

BrentGMATPrepNow · Answer

Here's an approach that doesn't involve modular arithmetic....

If we recognize that 82 is 1 greater than 81 (aka  3^4 ), then we might see that 82 is a divisor of  3^{32} - 1 
Here's why:

3^{32} - 1 = (3^{16} + 1)(3^{16} - 1) 
 = (3^{16} + 1)(3^8 + 1)(3^8 - 1) 
 = (3^{16} + 1)(3^8 + 1)(3^4 - 1)(3^4 + 1) 
 = (3^{16} + 1)(3^8 + 1)(3^4 - 1)(82)

This tells us that  (3^{32} - 1)  is a multiple of 82...
...and this means  3^{32}  is   1  greater than a multiple of 82 , which means we'll get a remainder of  1  when we divide  3^{32}  by 82

Answer: B

Cheers,
Brent

jamiedimonn · Answer

@Bkarishma,  Bunuel  or any other expert could you please explain why ans is B...

i did (3^4)^8 which makes it 81/82 which tells us that quotient is 0 and remainder is 81 making it answer E BUT WHY ANS IS B??

Krunaal · Answer

We are dividing  81^8  by  82  and  not   81^1  by  82 ,

When we divide  81^1  by  82 , we get  (-1)^1  =  -1  but as remainder cannot be negative we add divisor  82   to it =>  -1 + 82 = 81  thus  81  becomes the remainder.

But when we divide  81^8  by  82 , we get  (-1)^8 = 1 , and  1  can be a remainder.

Here's more on the concept of negative remainders =>  All about negative remainders on GMAT

What is the remainder when 3^32 is divided by 82?

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What is the remainder when 3^32 is divided by 82?

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