## Remainder

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### Remainder

by didieravoaka » Thu Mar 10, 2016 2:17 pm

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by [email protected] » Thu Mar 10, 2016 4:53 pm
Hi didieravoaka,

This question is ultimately about pattern-matching (and paying attention to the 'units digit' of each calculation). The GMAT won't expect you to calculate 3^19, so let's do some easier calculations instead and look for a pattern:

3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81

3^5 = 243

Notice how the units digit is now following a pattern: 3 - 9 - 7 - 1

This pattern will repeat with every 4 calculations that you perform, so you can deduce:

3^19 will be 4 "groups" of (3 - 9 - 7 - 1) and then a 3, a 9 and finally a 7

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by [email protected] » Thu Mar 17, 2016 9:45 pm
Another idea here:

3Â¹â�¹ =

(3â�´)â�´ * 3Â³

Since 3â�´ = 81, we know that (3â�´)â�´ = 81â�´ = something that ends in 1. So we have

(something that ends in 1) * 3Â³ =>

(something that ends in 1) * 27 =>

something that ends in 7

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by ceilidh.erickson » Fri Mar 18, 2016 8:48 am
First, we have to translate. Any time a question asks "What is the remainder when X is divided by 10?" it's really asking for the UNITS DIGIT.

For example, when 546 is divided by 10 --> 54 remainder 6
when 7247 is divided by 10 --> 724 remainder 7

As Rich showed, units digits of powers will form certain patterns. Establish that pattern, and you can easily find the answer.

Because powers of 3 repeat every 4 powers, then any time 3 is raised to a power that's a multiple of 4, the units digit will be 1. So we know that 3^16 will have a units digit of 1, and we can establish the pattern after that:

3^16 --> 1
3^17 --> 3
3^18 --> 9
3^19 --> 7
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by ceilidh.erickson » Fri Mar 18, 2016 8:50 am
Here are a few more examples of "what is the remainder when divided by 10" used as code for "what is the units digit":
https://www.beatthegmat.com/if-r-s-and-t ... tml#548713
https://www.beatthegmat.com/if-n-and-m-a ... tml#544266
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by ceilidh.erickson » Fri Mar 18, 2016 8:51 am
Ceilidh Erickson
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