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

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by Brent@GMATPrepNow » Sun Feb 23, 2020 8:51 am

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

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

Answer: B
Source: www.gmatprepnow.com
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Re: What is the remainder when 3^32 is divided by 82?

by Brent@GMATPrepNow » Mon Feb 24, 2020 6:59 am
Brent@GMATPrepNow wrote: ↑
Sun Feb 23, 2020 8:51 am
 What is the remainder when 3^32 is divided by 82?

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

Answer: B
Source: www.gmatprepnow.com
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
Brent Hanneson - Creator of GMATPrepNow.com
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