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# Modular Arithmetic Help

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1. When 310 is divided by a two-digit positive integer K, the remainder is 37. What is the sum of all possible values of K?

2. N is a positive integer. When N is divided by 3, the remainder is 2. When N is divided by 4, the remainder is 3. What is the remainder when N is divided by 12?

EDIT: TYPO!!!

It's When N is divided by 3, the remainder is 2

3. The zoo transferred 131 birds to the safari park over the course of many days.

The zookeeper planned to send the same number of birds each day, except that there might be fewer on the last day when he would send whatever was left. However, he mistakenly thought that there were 113 birds in total instead of 131, which caused the transfer to take 3 more days than he anticipated. Interestingly, the number of birds transferred during the last day was the same as he had originally planned.

How many birds were transferred during the last day?

4. Duncan found a set of positive integers less than 50 such that no two of them sum to a multiple of 7. What is the largest number of integers in Duncan's collection?

Jan 10, 2019
edited by sudsw12  Jan 11, 2019

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2. Rewriting this, we have: $$3x+3=4y+3, \frac{3x}{4}=y.$$ Now, plugging a value of $$x$$ that will leave $$y$$ are a non-negative integer, is $$x=4.$$ Answer is 3? I might be wrong...

Jan 10, 2019
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Hmm... that answer's wrong. Does anyone know why?

sudsw12  Jan 11, 2019
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2. N is a positive integer. When N is divided by 3, the remainder is 3. When N is divided by 4, the remainder is 3. What is the remainder when N is divided by 12?

If some number is divided by 3 how can the remainder be 3?

Jan 10, 2019
edited by Melody  Jan 10, 2019
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LOL!!!.....not possible.....!!!!

CPhill  Jan 10, 2019
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Whoops! I fixed that. Also, thanks CPhill for number 1.

sudsw12  Jan 11, 2019
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1. When 310 is divided by a two-digit positive integer K, the remainder is 37. What is the sum of all possible values of K?

310 - 37 =  273

273 = 3  * 7 * 13

310 / 39 =  7(39) + 37

310 / 91 = 3(91) + 37

Sum of all possible K's  =   39 + 91  =  130

Jan 10, 2019
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Thanks Chris,

I would not have thought to do that. :))

Melody  Jan 10, 2019