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 

18.33333 

13.75 

11 
56 

18.66667 

14 

11.2 
57 

19 

14.25 

11.4 
58 

19.33333 

14.5 

11.6 
59 

19.66667 

14.75 

11.8 
60 

20 

15 

12 
61 

20.33333 

15.25 

12.2 
62 

20.66667 

15.5 

12.4 
63 

21 

15.75 

12.6 
64 

21.33333 

16 

12.8 
65 

21.66667 

16.25 

13 
66 

22 

16.5 

13.2 
67 

22.33333 

16.75 

13.4 
68 

22.66667 

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?
Author & Contact
Insert name and contact information.
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Link(s) to resources, references, lesson plans, and/or other
materials
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