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Title
Remainder Shells

Problem Statement

Ken has between 3 and 100 shells in his collection. If he counts his shells 3 at a time, he has 2 left over. If he counts them 4 at a time, he has 2 left over. If Ken counts them 5 at a time, he still has 2 left over. How many shells does Ken have in his shell collection?

Problem setup

Ken is visiting peaceful Tybee Island.  He has a collection of shells.  If the shells are divided amongst three people, there are two remaining.  If the shells are divided amongst four and five persons, there are still two shells remaining.  How many shells does Ken have in his collection? Ken has between 3 and 100 shells in his collection.

Plans to Solve/Investigate the Problem

Strategy 1:

I have become enamored with the use of Excel to compile large amounts of data quickly in solving word problems (thanks Michelle Houston!).  I compiled a spreadsheet like this:

 by 3 by 4 by 5 3 1 0.75 0.6 4 1.333333 1 0.8 5 1.666667 1.25 1 6 2 1.5 1.2 7 2.333333 1.75 1.4 8 2.666667 2 1.6 9 3 2.25 1.8 10 3.333333 2.5 2 11 3.666667 2.75 2.2 12 4 3 2.4 13 4.333333 3.25 2.6 14 4.666667 3.5 2.8 15 5 3.75 3 16 5.333333 4 3.2 17 5.666667 4.25 3.4 18 6 4.5 3.6

Data was compiled for the numbers 3 through 100.  At the completion of the spreadsheet, I had to consider, what do I do with what I have?

Strategy #2:

I considered the use of the least common multiple of three, four and five.

Investigation/Exploration of the Problem

The least common multiple of three, four, and five is 60 (5 * 4 * 3 = 60).  Now, IF 60 is the least common multiple of three, four, and five AND IF there are two shells remaining if the shells are counted out three, four, and five at a time, THEN the least common factor, 60, plus 2, total of 62, should result in two shells left over when the shells are divided three, four and five at a time.

Let's check!

62 ÷ 5 = 12 with a remainder of 2

62 ÷ 4 = 15 with a remainder of 2

62 ÷ 3 = 20 with a remainder of 2

Now, what do I do with the data in my spreadsheet?

62 ÷ 5 = 12 with a remainder of 2 = 12.4

62 ÷ 4 = 15 with a remainder of 2 = 15.5

62 ÷ 3 = 20 with a remainder of 2 = 20.6666

Does a row exist in my spreadsheet where all three of these numbers occur ?  Yes, for the number 62!

 55 18.3333 13.75 11 56 18.6667 14 11.2 57 19 14.25 11.4 58 19.3333 14.5 11.6 59 19.6667 14.75 11.8 60 20 15 12 61 20.3333 15.25 12.2 62 20.6667 15.5 12.4 63 21 15.75 12.6 64 21.3333 16 12.8 65 21.6667 16.25 13 66 22 16.5 13.2 67 22.3333 16.75 13.4 68 22.6667 17 13.6 69 23 17.25 13.8

Extensions of the Problem

Would you approach the problem differently if the number of shells could be greater than 100?

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