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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 had less than 100 blocks which equaled 5 * n +1, 4 * n + 1, and 9 * n.

Plans to Solve/Investigate the Problem

At first I tried the guess and check method to determine a number that would satisfy the first 2 requirements, but then I remembered the third criteria.  So I looked at the products of 9.

Investigation/Exploration of the Problem

My first guesses were 96 and 71, but they did not = 4 * n + 1.  After I listed the multiples of 9 through 9 * 11, I noticed that 81 was divisible by both 5 and 4 with each leaving a remainder of 1.  81 is the answer!

Extensions of the Problem

Suppose that all the conditions about Jeff's blocks remain except that he has fewer than 1000 blocks. How many blocks might Jeff have?

I tried multiplying the multiples of 9 (through 9 * 11) by 2, but that wouldn't work since if you divided the numbers in the new list by 5 or 4, you would not get 1 as a remainder.  The same if I multiplied by 3, 4, 5, 6, 8, 10.  Then I tried multiplying by 9, but, again, only 81 satisfied all of the requirements.  Then I made a spread sheet to list all multiples of 9 until the numbers reached over 1000.  5 * n + 1 would have either a 6 or a 1 in the ones place.  I need only an odd number in the ones place, so my answers must end in 1 to satisfy that criteria.  Multiples of 4 plus 1 would have either 1, 5, 9, 3, 7 in the ones place, so to satisfy both 5 * n + 1 and 4 * n + 1, the answers must have a 1 in the ones digit.  After eliminating all other numbers on the spread sheet, I was left with 81, 171, 261, 351, 441, 531, 621, 711, 801, 891, and 981.  Next I checked each to see if it was divisible by 4 with a remainder of 1.  That eliminated 171, 351, 531, 711, and 891.

Jeff could have 81, 261, 441, 621, 801, or 981 blocks.

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