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 Write-up

Title
Blocked Off

Problem Statement
Jeff had fewer than 100 blocks. When he made five equal rows, he had one left over; with four equal rows, he had one left over; and with nine equal rows, he had none left over. How many blocks did Jeff have?

Problem setup

Jeff is making arrays with less than 100 blocks.  In some cases, he has blocks left in his hand when he arranges the clocks in rows.  In other cases, he is able to fit every block into the array.  The goal is to find out how many blocks Jeff is working with anyway?

Plans to Solve/Investigate the Problem

The value of blocks Jeff has is a multiple of 9 since with an array with rows of 9 blocks, he has no left-overs.

I need to find a multiple 5 and also a multiple of 4 that will be a value of 1 away from a multiple of 9.  This might be a common multiple of 5 and 4.

Investigation/Exploration of the Problem

Multiples of 9 = 9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99.  We can go no further since Jeff has fewer than 100 blocks.

Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96.  The next multiple of 4 would be 100, too large for our example.  I need to keep in mind that we are looking for a multiple of 4 that is 1 away from a multiple of 9.

Multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95; again 100 is too large for our example.  And again we are looking for a multiple of 5 that is 1 away from a multiple of 9.

Common multiples of 4 and 5 = 20, 40, 60, 80.  100 is too large for our problem situation.

Multiples of 4 which are 1 away from a multiple of 9 = 8, 44, 64, and 80.

Multiples of 5 which are 1 away from a multiple of 9 = 10, 35, 55, and 80.

Only one common multiple of 4 and 5 can be found in both of the individual lists...80.

Our multiple of 4 and 5 which is 1 away from a multiple of 9 is 80.  The multiple of 9 we are referring to is 81.  Let's show it in a picture.

9s                                                             5s                                             4s

*********                                        *****        *****                       ****       ****

*********                                        *****        *****                       ****       ****

*********                                        *****        *****                       ****       ****

*********                                        *****        *****                       ****       ****

*********                                        *****        *****                       ****       ****

*********                                        *****        *****                       ****       ****

*********                                        *****        *****                       ****       ****

*********                                        *****        *****                       ****       ****

*********                                                     *                                    ****       ****

81 with                                                   80 with one                             ****       ****

none left                                                    left over                                           *

80 with one

left over

Jeff has 81 blocks he is working with.

Extensions of the Problem

Can you find a different number of blocks Jeff can use in his work that would allow him to make rows of 9 with no left-overs and rows of  4, 5, and 3 with 1 left-over block for each case?  What would that number be?

Author & Contact
rrogers@rockdale.k12.ga.us