## A Multiplication Based Logic Puzzle

### 810 Stick and Stone

Print the puzzles or type the solution on this excel file: 10-factors 807-814

One of my education professors taught that you can teach any concept with a picture book.

I recently read the book, Stick and Stone, to a class of 6th graders. Yes, 6th graders. You can get away with reading something way below grade level if you tell them before you start reading that you will use the book to introduce them to something that is definitely NOT below grade level. The first few pages of the book are shared by its publisher here:

As you can see, those first few pages equate stone as a zero and stick as a lonely number one.

The middle part of the book teaches about synergizing, working together to make life good and helping each other through tough times.

By the end of the book Stick and Stone know how to work very well together, “Stick, Stone. Together again. Stick, Stone. A perfect ten.”

The book pretty much ends there, but making a perfect ten is only the beginning of what these two characters can do together. I used this book to teach the class not only about getting along and working together, but also about base 2, or binary as it is also called. Every counting number we know can be represented by using just 1’s and 0’s. I wrote on the board the numbers from 1 to 16 and represented the first few of those numbers in base 2. Then I invited class members to come up with how to write the rest of the numbers in base 2. Some students caught on immediately while the others were able to learn how to do it by watching their classmates and listening to them. Eventually with at least 12 different student’s inputs, we came up with a chart that looked something like this:

Notice that the numbers from 9 to 15 are just 1000 plus the numbers directly across from them in the first column.

Some of the sixth grade students had already heard of binary, so I showed them a little more about base 2: I wrote a bunch of 1’s and 0’s “off the top of my head” onto the board and added the headings to show place values: 1’s place, 2’s place, etc.

Then I told them to sum up the place values that contained a one:

Stick and Stone are the main two characters, but the book has one other character, Pinecone. At first Pinecone bullied Stone, but after Stick stood up to him, the three of them were eventually able to become friends. You might enjoy finding out more about Pinecone by listening to Sean Anderson read the entire book to his children, one of which seems to really enjoy numbers.

If you used a unique symbol to represent Pinecone, it could look like a 2. Then you also could use the symbols 0, 1, and 2 to represent every counting number in base 3. That’s another concept the picture book Stick and Stone could be used to introduce!

To make a chart for base 3, start with these 3 columns of numbers with 3 numbers in each:

Since this is base 3, where should 10 and 100 go? The bottom of the first column and the bottom of the third column both MUST look like a power of 10. The rest of the chart is easy to fill out. Notice the 1 and 2 look exactly the same in base 10 and base 3. Also since 4 = 3 + 1, 5 = 3 + 2, and 6 = 3 + 3, we can easily fill in the 2nd column. Two more addition facts will finish the third column: 7 = 6 + 1, and 8 = 6 + 2.

Now add what you learned about 4, 5, 6, 7, 8, and 9 to column 1 and put the numbers 10 – 18 in the base 10 second column and numbers 19 – 27 in the base 10 third column. Again the bottom of the first column and the bottom of the third column both MUST look like a power of 10, so we now know where to put 1000.

To fill in the rest of the chart simply add 100 to the base 3 numbers in column 1 to get the the base 3 numbers in column 2. Then add 200 to the base 3 numbers in column 1 to get the remaining base 3 numbers in column 3.

You could do this process again to determine the first 81 counting numbers in base 3 with 81 being represented by 10000.

For base 4, you could do something similar with 4 columns. However, for counting in bases 4, 5, 6, 7, 8, and 9 I would suggest using the very versatile hundred chart. You can give instructions without even mentioning the concept of differing bases. For example, cross out every number on the hundred chart that has 7, 8, or 9 as one or more of its digits. Can you tell even before you get started how many numbers will get crossed out? (100 – 7²) What pattern do the cross-outs make? If you arrange the remaining numbers in order from smallest to largest, then you will have the first 49 numbers represented in base 7. With a minimal amount of cutting and taping you could have a “hundred” chart in base 7. Easy peasy.

This excel file not only has several puzzles, including today’s, but also a hundred chart and even a thousand chart because I know some of you might want to play with 3-digit numbers, too.

Now let me tell you a little bit about the number 810:

• 810 is a composite number.
• Prime factorization: 810 = 2 x 3 x 3 x 3 x 3 x 5, which can be written 810 = 2 x $3^4$ x 5
• The exponents in the prime factorization are 1, 4 and 1. Adding one to each and multiplying we get (1 + 1)(4 + 1)(1 + 1) = 2 x 5 x 2 = 20. Therefore 810 has exactly 20 factors.
• Factors of 810: 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 81, 90, 135, 162, 270, 405, 810
• Factor pairs: 810 = 1 x 810, 2 x 405, 3 x 270, 5 x 162, 6 x 135, 9 x 90, 10 x 81, 15 x 54, 18 x 45 or 27 x 30
• Taking the factor pair with the largest square number factor, we get √810 = (√81)(√10) = 9√10 ≈ 28.4604989.

Since 810 has so many factors, it has MANY possible factor trees. If most people made a factor tree for 810, they would probably start with 81 × 10 or 9 x 90. NOT ME! Here are two less-often-used factor trees for 810:

Finally, here is an easy way to express 810 is in a different base:

1. Make a cake in which you divide 810 by the base number repeatedly, keeping track of the remainders, including zero, as you go.
2. Keep dividing until the number at the top of the cake is 0.
3. List the remainders in order from top to bottom and indicate the base you used to do the division.
4. This method is illustrated for BASE 2 and BASE 3 below:

That’s all pretty good work for a stone, a stick, and a pine cone!

By the way, using that method will also produce the following results:

• 810 is 30222 BASE 4
• 810 is 11220 BASE 5
• 810 is 3430 BASE 6 and so forth.

### 807 and Level 1

What can I say about the number 807?

807 is palindrome 151 in BASE 26 because 1(26²) + 5(26) + 1(1) = 807.

Anything else? Well, I can figure out a few other things because 807’s has two prime factors, 3 and 269:

We can write ANY number (unless it’s a power of 2) as the sum of consecutive numbers in at least one way. 807 has three different ways to do that:

• 403 + 404 = 807 because 807 isn’t divisible by 2.
• 268 + 269 + 270 = 807 because it is divisible by 3.
• 132 + 133 + 134 + 135 + 136 + 137 = 807 since it is divisible by 3 but not by 6.

I know that one of 807’s factors, 269, is a hypotenuse of a Pythagorean triple, so 807 is also. Thus. . .

• (3·69)² + (3·260)² = (3·269)², or in other words, 207² + 780² = 807²

Since 807 has two odd sets of factor pairs, I know that 807 can be written as the difference of two squares two different ways:

• 136² – 133² = 807
• 404² – 403² = 807

I don’t usually do this, but today’s puzzle has something in common with 807. Can you tell what it is?

Print the puzzles or type the solution on this excel file: 10-factors 807-814

• 807 is a composite number.
• Prime factorization: 807 = 3 x 269
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 807 has exactly 4 factors.
• Factors of 807: 1, 3, 269, 807
• Factor pairs: 807 = 1 x 807 or 3 x 269
• 807 has no square factors that allow its square root to be simplified. √807 ≈ 28.4077454

### 572 and Level 1

72 can be evenly divided by 4 so 572 is divisible by 4.

5 – 7 + 2 = 0, which can be evenly divided by 11, so 11 is a factor of 572.

572 is the hypotenuse of the Pythagorean triple 220-528-572. Which factor of 572 is the greatest common factor of those three numbers?

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-03

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• 572 is a composite number.
• Prime factorization: 572 = 2 x 2 x 11 x 13, which can be written 572 = (2^2) x 11 x 13
• The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 572 has exactly 12 factors.
• Factors of 572: 1, 2, 4, 11, 13, 22, 26, 44, 52, 143, 286, 572
• Factor pairs: 572 = 1 x 572, 2 x 286, 4 x 143, 11 x 52, 13 x 44, or 22 x 26
• Taking the factor pair with the largest square number factor, we get √572 = (√4)(√143) = 2√143 ≈ 23.91652

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### 564 and Level 6

Consecutive primes 281 and 283 add up to 564.

564 is made from three consecutive numbers so it can be evenly divided by 3. If the middle number is divisible by 3, then a number made from three consecutive numbers will also be divisible by 9. Is 564 divisible by 9? Why or why not?

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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• 564 is a composite number.
• Prime factorization: 564 = 2 x 2 x 3 x 47, which can be written 564 = (2^2) x 3 x 47
• The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 564 has exactly 12 factors.
• Factors of 564: 1, 2, 3, 4, 6, 12, 47, 94, 141, 188, 282, 564
• Factor pairs: 564 = 1 x 564, 2 x 282, 3 x 188, 4 x 141, 6 x 94, or 12 x 47
• Taking the factor pair with the largest square number factor, we get √564 = (√4)(√141) = 2√141 ≈ 23.74868

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### 563 and Level 5

563 is prime, and it is the sum of all the prime numbers from 5 to 67.

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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• 563 is a prime number.
• Prime factorization: 563 is prime and cannot be factored.
• The exponent of prime number 563 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 563 has exactly 2 factors.
• Factors of 563: 1, 563
• Factor pairs: 563 = 1 x 563
• 563 has no square factors that allow its square root to be simplified. √563 ≈ 23.727621

How do we know that 563 is a prime number? If 563 were not a prime number, then it would be divisible by at least one prime number less than or equal to √563 ≈ 23.7. Since 563 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 563 is a prime number.

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### 562 and Level 4

562 is the sum of all the prime numbers from 23 to 71.

562 is also the hypotenuse of the Pythagorean triple 320-462-562. Which factor of 562 is the greatest common factor of those three numbers?

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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• 562 is a composite number.
• Prime factorization: 562 = 2 x 281
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 562 has exactly 4 factors.
• Factors of 562: 1, 2, 281, 562
• Factor pairs: 562 = 1 x 562 or 2 x 281
• 562 has no square factors that allow its square root to be simplified. √562 ≈ 23.7065

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### 560 and Level 3

560 is the hypotenuse of the Pythagorean triple 336-448-560. What is the greatest common factor of those three numbers?

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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• 560 is a composite number.
• Prime factorization: 560 = 2 x 2 x 2 x 2 x 5 x 7, which can be written 560 = (2^4) x 5 x 7
• The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 560 has exactly 20 factors.
• Factors of 560: 1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 28, 35, 40, 56, 70, 80, 112, 140, 280, 560
• Factor pairs: 560 = 1 x 560, 2 x 280, 4 x 140, 5 x 112, 7 x 80, 8 x 70, 10 x 56, 14 x 40, 16 x 35 or 20 x 28
• Taking the factor pair with the largest square number factor, we get √560 = (√16)(√35) = 4√35 ≈ 23.664319

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.