anonymous
  • anonymous
When positive integer N is divided by 167, the remainder is 35, and when positive integer K is divided by 167, the remainder is 17. What is the remainder when 2N+K is divided by 167?
Mathematics
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
If \(\dfrac{N}{167}\) gives a remainder of \(35\), then \(\dfrac{2N}{167}\) will have a remainder of \(70\) (twice \(35\)). Then if \(\dfrac{K}{167}\) has remainder \(17\), adding \(\dfrac{2N+K}{167}\) would add the remainders as well to get \(\dfrac{2(35)+17}{167}=\dfrac{87}{167}\). Had the sum of the remainders been larger than \(167\), you would have to make a slight adjustment. For example, if you had a total remainder of \(\dfrac{168}{167}\), then the numerator is \(1\) greater than \(167\), which means your remainder would have been \(1\). Notice that any remainders that differ by \(167\) would actually be equivalent. You can learn more here if you so choose. https://en.wikipedia.org/wiki/Modular_arithmetic#Remainders

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