## A Multiplication Based Logic Puzzle

### 816 and Level 2

Eight is half of sixteen, so 816 is divisible by 6. You probably weren’t expecting that divisibility rule, but it’s true.

816 can also be easily divided by 2, 4, and 8. How many factors does 816 have in all? Plenty! Scroll down past the puzzle and see!

Print the puzzles or type the solution on this excel file: 12 factors 815-820

• 816 is a composite number.
• Prime factorization: 816 = 2 x 2 x 2 x 2 x 3 x 17, which can be written 816 = 2⁴ x 3 x 17
• 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 816 has exactly 20 factors.
• Factors of 816: 1, 2, 3, 4, 6, 8, 12, 16, 17, 24, 34, 48, 51, 68, 102, 136, 204, 272, 408, 816
• Factor pairs: 816 = 1 x 816, 2 x 408, 3 x 272, 4 x 204, 6 x 136, 8 x 102, 12 x 68, 16 x 51, 17 x 48 or 24 x 34
• Taking the factor pair with the largest square number factor, we get √816 = (√16)(√51) = 4√51 ≈ 28.5657

Since 17 is one of its factors, 816 is the hypotenuse of a Pythagorean triple:

• 384-720-816 which is 48 times 8-15-17

816 is repdigit OO in base 33 (O is 24 base 10). That is true because

• 24(33¹) + 24(33º) = 24(33¹ + 33º) = 24(33 + 1) = 24 × 34 = 816

816 is the sum of the sixteen prime numbers from 19 to 83:

• 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 816

Coincidentally, 816 is also the sixteenth tetrahedral number.

That’s because 16(16 + 1)(16 + 2)/6 = 816, which is a fast way to compute it. Here’s what it means to be the 16th tetrahedral number:

### 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.

### 809 Palindromes, Factors, Whoop-de-doo

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

Normally I would tell you that 809 is a palindrome in two different bases:

• 676 in BASE 11 because 6(121) + 7(11) + 6(1) = 809
• 575 in BASE 12 because 5(144) + 7(12) +5(1) = 809

But whoop-de-doo, all that really means is that (x – 11) is a factor of 6x² + 7x – 803, and (x – 12) is a factor of 5x² + 7x – 804.

Isn’t it just as exciting that

• (x – 2) is a factor of x^9 + x^8 + x^5 + x³ – 808 because 809 is 1100101001 in BASE 2?
• (x – 3) is a factor of x^6 + 2x³ + 2x² +2x – 807 because 809 is 1002222 in BASE 3?
• (x – 4) is a factor of 3x^4 + 2x² + 2x – 808 because 809 is 30221 in BASE 4?
• (x – 5) is a factor of x^4 + x³ + 2x² + x – 805 because 809 is 11214 in BASE 5?

Notice that the last number in each of those polynomials is divisible by the BASE number.

Palindromes NEVER end in zero so the polynomials they produce will NEVER end in the original base 10 number.

So are palindromes really so special? Today I am much more excited that figuring out what a number is in another base can give us a factor of a corresponding polynomial!

How do I know what those polynomials are? Let me use 809 in BASE 6 as an example:

Since 809 is 3425 in BASE 6, I know that

• 3(6³) + 4(6²) + 2(6¹) + 5(6º) = 809
• 3(216) + 4(36) + 2(6) + 5(1) – 809 = 0
• so 3(216) + 4(36) + 2(6) – 804 = 0
• thus (x – 6) is a factor of 3x³ + 4x² + 2x – 804 because of the factor theorem.

If I told you what 809 is in Bases 7, 8, 9, and 10 would you be able to write the corresponding polynomials that are divisible by (x – 7), (x – 8), (x – 9), and (x – 10) respectively?

• 2234 in BASE 7
• 1451 in BASE 8
• 1088 in BASE 9
• 809 in BASE 10

Scroll down past 809’s factoring information to see if you found the correct polynomials.

—————–

• 809 is a prime number.
• Prime factorization: 809 is prime and cannot be factored.
• The exponent of prime number 809 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 809 has exactly 2 factors.
• Factors of 809: 1, 809
• Factor pairs: 809 = 1 x 809
• 809 has no square factors that allow its square root to be simplified. √809 ≈ 28.4429253066558.

How do we know that 809 is a prime number? If 809 were not a prime number, then it would be divisible by at least one prime number less than or equal to √809 ≈ 28.4. Since 809 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 809 is a prime number.

Here’s another way we know that 809 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 28² + 5² = 809 with 28 and 5 having no common prime factors, 809 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √809 ≈ 28.4. Since 809 is not divisible by 5, 13, or 17, we know that 809 is a prime number.

——————–

Were you able to find those polynomials from knowing what 809 is in other bases? Check your work with the answers below:

• 2234 Base 7 tells us (x – 7) is a factor of 2x³ + 2x² + 3x – 805
• 1451 Base 8 tells us (x – 8) is a factor of x³ + 4x² + 5x – 808
• 1088 Base 9 tells us (x – 9) is a factor of x³ + 8x – 801
• 809 Base 10 tells us (x – 10) is a factor of 8x² – 800

If you’ve made it this far, even if I’ve made you feel a little dizzy, you’ve done GREAT! Keep up the good work!

### 803 From Top to Bottom

8 – 0 + 3 = 11, so 803 is divisible by 11.

• 803 is a composite number.
• Prime factorization: 803 = 11 x 73
• 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 803 has exactly 4 factors.
• Factors of 803: 1, 11, 73, 803
• Factor pairs: 803 = 1 x 803 or 11 x 73
• 803 has no square factors that allow its square root to be simplified. √803 ≈ 3372546

You can solve today’s Level 3 puzzle by starting at the top of the first column, finding the factors of the clues and writing them in the appropriate cells. Then continue to go down that same column, cell by cell, finding factors and writing them down until you reach the bottom. Make sure that both the first column and the top row have each number from 1 to 10 written in them.

Print the puzzles or type the solution on this excel file: 10-factors 801-806

Here’s a few more facts about the number 803:

803 is the hypotenuse of a Pythagorean triple:

• 528-605-803 which is 11 times another Pythagorean triple: 48-55-73

803 is the sum of three squares six different ways:

• 27² + 7² + 5² = 803
• 25² + 13² + 3² = 803
• 23² + 15² + 7² = 803
• 21² + 19² + 1² = 803
• 19² + 19² + 9² = 803
• 17² + 17² + 15² = 803

803 is the sum of consecutive prime numbers three different ways. Prime factor 11 is not in any of those ways, but prime factor 73 is in two of them.

• 263 + 269 + 271 = 803, that’s 3 consecutive primes.
• 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 = 803, that’s 9 consecutive primes.
• 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 = 803, that’s 13 consecutive primes.

803 is a palindrome in two bases. Why are the numbers similar in these two palindromes?

• 30203 BASE 4 because 3(256) + 0(64) + 2(16) + 0(4) + 3(1) = 803
• 323 BASE 16 because 3(16²) + 2(16) + 3(1) = 803

### 802 Pi Day at Smith’s

In the United States tomorrow’s date is written 3-14. Because 3.14 is a famous approximation for π (pi), people all over the country will eat pie to celebrate Pi Day. This afternoon I took a picture of this sign and the pie display at my local Smith’s Food and Drug.

I took that picture right when I walked into the store, but there were no pies on display for National Pi Day.

About 15 minutes later I returned to the display to take another picture. Now there were pies on the table! I told a salesperson who I think worked on the display that I was going to take a picture and put it on my blog. She asked what kind of a blog I wrote. I told her a math blog. She looked puzzled and asked why I would want to put a picture of pies on a math blog. Then she turned around, looked at the display, and said something like, “Oh, now I get it, the number pi.”

How do you choose between apple, cherry, or peach pie? It’s much easier if you choose two and then you can get a free 8 oz. Cool Whip, too. Yummy.

If by chance you prefer pizza pi, here’s a thought from twitter that is often repeated in March:

And here’s some original artwork that displays pi in a way I had never thought of before:

BREAKING: secret of Pi revealed #PiDay pic.twitter.com/Ao8BQp31jd

You can also look here for a million digits of pi.

But pi is not the only interesting number in the world. Every number has its own curiosities. Let me tell you some reasons to get excited about the number 802:

802 is the sum of two squares:

• 21² + 19² = 802

So 802 is the hypotenuse of a Pythagorean triple:

• 80-798-802, which is 2 times another triple: 40-399-401.

It also means something else: Since odd numbers 21 and 19 have no common prime factors, 802 can be evenly divide by 2. Duh. . ., but it also means that unless 802 is also divisible by 5, 13, or 17, its only factors will be 2 and a prime number! Why are those three numbers the only ones I care about? Because they are the only prime number Pythagorean triple hypotenuses less than √802 ≈ 28.3.

Guess what? 5, 13, and 17 do not divide evenly into 802, so 802 is the product of 2 and a prime number which happens to be 401.

• 802 is a composite number.
• Prime factorization: 802 = 2 x 401
• The exponents in the prime factorization are 1 and 1. Adding one to each exponent and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 802 has exactly 4 factors.
• Factors of 802: 1, 2, 401, 802
• Factor pairs: 802 = 1 x 802 or 2 x 401
• 802 has no square factors that allow its square root to be simplified. √802 ≈ 28.3196045

Today’s puzzle is number 802 to distinguish it from every other puzzle I’ve made. Writing the numbers 1 – 10 in both the top row and the first column so that the factors and the clues work together as a multiplication table is as easy as pie!

Print the puzzles or type the solution on this excel file: 10-factors 801-806

And here is a little more about the number 802:

802 is the sum of 8 consecutive prime numbers:

• 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 = 802

802 can also be written as the sum of three squares three different ways:

• 28² + 3² + 3² = 802
• 27² + 8² + 3² = 802
• 24² + 15² + 1² = 802

802 is also a palindrome in two other bases:

• 414 BASE 14 because 4(196) + 1(14) + 4(1) = 802
• 202 BASE 20 because 2(400) + 0(20) + 2(1) = 802

### 796 and a Valentine

Here’s a puzzle you can print, cut out, and give as a valentine:

You may know the divisibility rules for these powers of two:

• If the last digit of a number is divisible by 2, the whole number is divisible by 2.
• If the last two digits are divisible by 4, the whole number is divisible by 4.
• If the last three digits are divisible by 8, the whole number is divisible by 8.

But I’m going to apply some other time-saving but possibly more confusing divisibility rules to the number 796:

• 796 is divisible by 2 because 6 is an even number.
• 796 is divisible by 4 because even number 6 is NOT divisible by 4, and 9 is an odd number.
• 796 is NOT divisible by 8 because 96 is divisible by 8, and 7 is an odd number.

Because 796 is divisible by 4 but not by 8, it can be written as the sum of 8 consecutive numbers:

• 96 + 97 + 98 + 99 + 100 + 101 + 102 + 103 = 796

796 is also the sum of all the prime numbers from 113 to 149:

• 113 + 127 + 131 + 137 + 139 + 149 = 796

Here is the factoring information for 796:

• 796 is a composite number.
• Prime factorization: 796 = 2 x 2 x 199, which can be written 796 = (2^2) x 199
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 796 has exactly 6 factors.
• Factors of 796: 1, 2, 4, 199, 398, 796
• Factor pairs: 796 = 1 x 796, 2 x 398, or 4 x 199
• Taking the factor pair with the largest square number factor, we get √796 = (√4)(√199) = 2√199 ≈ 28.21347.

### 789 and Level 2

789 consists of exactly three consecutive numbers so it is divisible by 3.

Print the puzzles or type the solution on this excel file: 10-factors-788-794

————————————–

• 789 is a composite number.
• Prime factorization: 789 = 3 x 263
• 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 789 has exactly 4 factors.
• Factors of 789: 1, 3, 263, 789
• Factor pairs: 789 = 1 x 789 or 3 x 263
• 789 has no square factors that allow its square root to be simplified. √789 ≈ 28.08914.

789 is the sum of consecutive prime numbers 2 different ways:

• 257 + 263 + 269 = 789 (that’s 3 consecutive primes.)
• 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 = 789 (that’s 21 consecutive primes!)

789 is the sum of three squares eight different ways:

• 28² + 2² + 1² = 789
• 26² + 8² + 7² = 789
• 25² + 10² + 8² = 789
• 23² + 16² + 2² = 789
• 23² + 14² + 8² = 789
• 22² + 17² + 4² = 789
• 22² + 16² + 7² = 789
• 20² + 17² + 10² = 789