A Multiplication Based Logic Puzzle

Archive for the ‘Mathematics’ Category

808 Happy Birthday, Justin!

Happy birthday to my good friend, Justin! He seems to always remember the birthday’s of everyone he knows, so this is how I am remembering his special day today. Justin is highly intelligent, thoughtful, and very friendly. I am confident he can solve this Level 6 puzzle that looks a little like a birthday cake.

This is my 808th post so I thought I would also make a factor cake for the number 808. It’s prime factor, 101, is at the top of the cake. Justin, I hope you live to be 101!

808 is a palindrome. That means it looks the same forwards and backwards. It is also a strobogrammatic number. That means it looks the same right side up or upside down.

ALL of the factors of 808 are also palindromes, and four of them are strobogrammatic numbers, too. Can you figure out which ones are both?

  • 808 is a composite number.
  • Prime factorization: 808 = 2 x 2 x 2 x 101, which can be written 808 = (2^3) x 101
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 808 has exactly 8 factors.
  • Factors of 808: 1, 2, 4, 8, 101, 202, 404, 808
  • Factor pairs: 808 = 1 x 808, 2 x 404, 4 x 202, or 8 x 101
  • Taking the factor pair with the largest square number factor, we get √808 = (√4)(√202) = 2√202 ≈ 428.425340807

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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?

  • 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

 

 

 

 

 

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:

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

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

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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

 

800 Which Pony Will Take Second Place?

Every 100 posts I summarize the amount of factors of the previous 100 numbers.

MANY of the numbers from 701 to 800 have FOUR factors, and any other number-of-factors doesn’t even come close. For this Horse Race, SECOND place is much more interesting as there are several lead changes. I’ve shorten the track so the second place number-of-factors can reach the finish line.

So go ahead, pick the number-of-factors pony you think will come in SECOND place. Your best bets are 2, 6, 8, 12, 16 OR the second row of 4 factors!

Make your selection, then click on the graphic below to see how your pony does!

800-horse-race-01

Now let me tell you a little bit about the number 800.

800-prime-factorization

  • 800 is a composite number.
  • Prime factorization: 800 = 2 x 2 x 2 x 2 x 2 x 5 x 5, which can be written 800 = (2^5) x (5^2)
  • The exponents in the prime factorization are 5 and 2. Adding one to each and multiplying we get (5 + 1)(2 + 1) = 6 x 3 = 18. Therefore 800 has exactly 18 factors.
  • Factors of 800: 1, 2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 80, 100, 160, 200, 400, 800
  • Factor pairs: 800 = 1 x 800, 2 x 400, 4 x 200, 5 x 160, 8 x 100, 10 x 80, 16 x 50, 20 x 40 or 25 x 32
  • Taking the factor pair with the largest square number factor, we get √800 = (√400)(√2) = 20√2 ≈ 28.28427.

800-factor-pairs

800 is the sum of four consecutive primes:

  • 193 + 197 + 199 + 211 = 800

800 is a palindrome in three different bases.

  • 2222 BASE 7 because 2(7^3) + 2(49) + 2(7) + 2(1) = 800 (This fact also means that x^3 + x² + x – 399 can be divided evenly by x – 7.)
  • 242 BASE 19 because 2(19²) + 4(19) + 2(1) = 800
  • PP BASE 31 (P is 25 base 10) because 25(31) + 25(1) = 800

800 is the sum of two squares two different ways:

  • 28² + 4² = 800
  • 20² + 20² = 800

That being true, it follows that 800 is the hypotenuse of two Pythagorean triples:

  • 480-640-800 which is 160 times 3-4-5
  • 224-768-800 which is 32 times 7-24-25

 

800 is also the sum of three squares:

  • 20² + 16² + 12² = 800

This chart summarizes the number of factors for the first 800 numbers and indicates that 39% of those numbers have square roots that can be simplified (reduced).

800-totals

In case you didn’t click on the Horse Race image before, here it is, no clicking required:

800 Horse Race

make science GIFs like this at MakeaGif

 

792 Number Facts and Factors of the Year 2017

Happy New Year, everybody!

So much of what I’ve included in this post is a bit difficult for elementary school students, but here is an area problem that they should be able to do:

2017-find-the-area

Perhaps you’ll recognize that as just another way to illustrate that 44² + 9² = 2017.

This area problem based on 33² + 28² + 12² = 2017 will be a little bit more challenging for students:

2017-area

2016 had more factors, positive and negative, than anybody could have imagined, but 2017 is a prime number year, so hopefully it will be filled with less drama.

I have a lot to say about the number 2017 with a little help from twitter.

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2017 has only TWO positive real factors, 1 and 2017, and only TWO negative real factors, -1 and -2017. Positive or negative, ALL the real possible factor pairs for 2017 are

  • 1 x 2017 and (-1) x (-2017).

2017 = 4(504) + 1. So 2017 is the sum of two squares. Which ones?

44² + 9² = 2017. That sum-of-squares number fact means that 2017 is the hypotenuse of a Pythagorean triple, specifically, 792-1855-2017. Since 2017 is a prime number, this triple is also a primitive.

2017-triple

Here’s how those numbers were calculated from the fact that  44² + 9² = 2017:

  • 2(44)(9) = 792
  • 44² – 9² = 1855
  • 44² + 9² = 2017

2017 is also the short leg in a rather monstrous primitive Pythagorean triple:

  • 2017² + 2,034,144² = 2,034,145²

Hmm. 44² + 9² = 2017 means we can find some COMPLEX factor pairs for 2017:

  • (44 + 9i)(44 – 9i) = 2017
  • (-44 + 9i)(-44 – 9i) = 2017
  • (9 + 44i)(9 – 44i) = 2017
  • (-9 + 44i)(-9 – 44i) = 2017

Who knows what all those COMPLEX factors will bring to the coming year? Each of them was derived from the fact that 44² + 9² = 2017.

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2017 is the sum of THREE squares three different ways:

  • 37² + 18² + 18²  = 2017
  • 33² + 28² + 12² = 2017
  • 30² + 26² + 21² = 2017

Since 37² = 35² + 12², 30² = 24² + 18², and 26² = 24² + 10², we can write 2017 as the sum of these squares, too:

  • 35² + 18² + 18² + 12² = 2017
  • 30² + 24² + 21² + 10² = 2017
  • 26² + 24² + 21² + 18² = 2017
  • 24² + 24² + 21² + 18² + 10² = 2017

Here’s more sums and/or differences of squares from twitter:

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This representation of 2017 is “two” powerful:

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2017 is a PALINDROME in bases 31, 32, and 36:

  • 232 BASE 31; note that 2(31²) + 3(31) + 2(1) = 2017
  • 1V1 BASE 32 (V is 31 base 10); note that 1(32²) + 31(32) + 1(1) = 2017
  • 1K1 BASE 36 (K is 20 base 10); note that 1(36²) + 20(36) + 1(1) = 2017

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Also Stetson.edu tells us this 2017 fact about the totient function: “2017 is a value of n for which φ(n) = φ(n-1) + φ(n-2).”

PROOF that 2017 is a prime number:

We can use these three facts to help us verify that 2017 is a prime number:

  1. 2017 = 4(504) + 1
  2. 44² + 9² = 2017
  3. √2017 ≈ 44.8998886

Since 2017 can be written as one more than a multiple of four, and 44 and 9 have no common prime factors, 2017 will be a prime number unless it is divisible by 5, 13, 17, 29, 37, or 41 (all the prime numbers less than √2017 ≈ 44.9 that have a remainder of one when divided by 4). That’s right, we ONLY have to divide 2017 by those SIX numbers to verify that it is prime: (Read here for why this is true.)

  • 2017 ÷ 5 = 403.4
  • 2017 ÷ 13 ≈ 155.15
  • 2017 ÷ 17 ≈ 118.65
  • 2017 ÷ 29 ≈ 69.55
  • 2017 ÷ 37 ≈ 54.51
  • 2017 ÷ 41 ≈ 49.20

We don’t get a whole number answer for any of those divisions, so 2017 is prime!

If you’re looking for more reasons to be interested in the number 2017, read David Radcliffe’s article:

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Here are some other lovely mathematical thoughts about 2017 that I found on twitter:

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In case you don’t know (and apparently a lot of people don’t know), 0! = 1, so this expression really does equal 2017.

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And finally, if you click on this next link, David Mitchell will explain the tessellation of the number 2017.

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Since this is my 792nd post, I’ll write a little bit about the number 792:

792 has 24 factors so I’ll include a few of its many possible factor trees:

792-factor-trees

Since 792 has so many factors, it is a leg in MANY Pythagorean triples. Here I list some factors that generate PRIMITIVE triples:

  • 2(396)(1): 792-156815-156817
  • 2(99)(4): 792-9785-9817
  • 2(44)(9): 792-1855-2017 (illustrated near the top of this post)
  • 2(36)(11): 792-1175-1417

792 is the sum of three squares four different ways including three ways that repeat squares:

  • 28² + 2² + 2² = 792
  • 26² + 10² + 4² = 792
  • 20² + 14² + 14² = 792
  • 18² + 18² + 12² = 792

792 is a palindrome in bases 32 and 35:

  • OO BASE 32 (O is 24 base 10); note that 24(32) + 24(1) = 792
  • MM BASE 35 (M is 22 base 10); note that 22(35) + 22(1) = 792

Finally here is the factoring information for the number 792:

  • 792 is a composite number.
  • Prime factorization: 792 = 2 x 2 x 2 x 3 x 3 x 11, which can be written 792 = (2^3) x (3^2) x 11
  • The exponents in the prime factorization are 3, 2 and 1. Adding one to each and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 x 3 x 2 = 24. Therefore 792 has exactly 24 factors.
  • Factors of 792: 1, 2, 3, 4, 6, 8, 9, 11, 12, 18, 22, 24, 33, 36, 44, 66, 72, 88, 99, 132, 198, 264, 396, 792
  • Factor pairs: 792 = 1 x 792, 2 x 396, 3 x 264, 4 x 198, 6 x 132, 8 x 99, 9 x 88, 11 x 72, 12 x 66, 18 x 44, 22 x 36 or 24 x 33

Taking the factor pair with the largest square number factor, we get √792 = (√36)(√22) = 6√22 ≈ 28.14249.

792-factor-pairs

780 is the 39th Triangular Number

  • 780 is a composite number.
  • Prime factorization: 780 = 2 x 2 x 3 x 5 x 13, which can be written 780 = (2^2) x 3 x 5 x 13
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 3 x 2 x 2 = 24. Therefore 780 has exactly 24 factors.
  • Factors of 780: 1, 2, 3, 4, 5, 6, 10, 12, 13, 15, 20, 26, 30, 39, 52, 60, 65, 78, 130, 156, 195, 260, 390, 780
  • Factor pairs: 780 = 1 x 780, 2 x 390, 3 x 260, 4 x 195, 5 x 156, 6 x 130, 10 x 78, 12 x 65, 13 x 60, 15 x 52, 20 x 39, or 26 x 30
  • Taking the factor pair with the largest square number factor, we get √780 = (√4)(√195) = 2√195 ≈ 27.92848.

780-factor-pairs

There are MANY ways to make factor trees for 780. Here are just three of them:

780 Factor Trees

Stetson.edu informs us that (7 + 5)(8 + 5)(0 + 5) = 780.

780 is the sum of consecutive prime numbers two different ways:

  • 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 = 780 (10 consecutive primes).
  • 191 + 193 + 197 + 199 = 780 (4 of my favorite consecutive primes).

26 and 30 are both exactly 2 numbers away from their average 28, so 780 can be written as the difference of two squares:

780 = 26 x 30 = (28 – 2)(28 + 2) = 28² – 2² = 784 – 4.

(It can be written as the difference of two squares three other ways, but I won’t list them here.)

Because 5 and 13 are both factors of 780, it is the hypotenuse of FOUR Pythagorean triangles making each of these equations true:

  • 192² + 756² = 780²
  • 300² + 720² = 780²
  • 396² + 672² = 780²
  • 468² + 624² = 780²

780 can be written as the sum of three perfect squares two different ways:

  • 26² + 10² + 2² = 780
  • 22² + 14² + 10² = 780

780 BASE 10 is palindrome QQ BASE 29. (Q is 26 in BASE 10.) Note that 26(29) + 26(1) = 780.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 = 780, making 780 the 39th triangular number. Since 20 is in the exact middle of that list of numbers, 780 is also the 20th hexagonal number.

Triangular numbers are interesting, but are they good for anything? Here’s one good thing:

Count the Terms of Sums Squared

There are 26 letters in the English alphabet. 39 variables would use the alphabet exactly one and one half times, but that’s okay because we can use upper case letters the second time around:

(a + b + c + . . . + x + y+ z + A + B + C + . . . + K + L + M)² has exactly 780 terms because it has 39 single variables, and 780 is the 39th triangular number.

780 is the 39th triangular number because 39⋅40/2 = 780.

A couple of months ago I saw a fascinating image on twitter. It is a square made with 1001 dots representing the numbers from -500 to +500 with zero in the exact middle. The triangular numbers are represented by the “+” pattern seen here. If we added about 600 more dots to the square, -780 and +780 would also lie on that +.

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Half of all triangular numbers are also hexagonal numbers. If there were enough dots, then -780 and +780 would also be among the dots forming the “-” pattern representing the hexagonal numbers. 780 is the 20th hexagonal number because 20(2⋅20-1) = 780.

 

How Lucky Can 777 Be?

  • 777 is a composite number.
  • Prime factorization: 777 = 3 x 7 x 37
  • The exponents in the prime factorization are 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 = 8. Therefore 777 has exactly 8 factors.
  • Factors of 777: 1, 3, 7, 21, 37, 111, 259, 777
  • Factor pairs: 777 = 1 x 777, 3 x 259, 7 x 111, or 21 x 37
  • 777 has no square factors that allow its square root to be simplified. 77727.8747197.

777-factor-pairs

Some people think that 7 is a lucky number. If that is true, then 777 should be even luckier.

Some numbers are lucky enough to be included in Multiplication Rhymes.ppt – mathval, a fun power point that helps students learn 12 multiplication facts, including these three that use Lucky Numbers:

  • 3 & 7 are always lucky numbers; 3 x 3 = 9 lives of a cat.
  • 3 & 7 are always lucky numbers; 3 x 7 = 21 lucky age.
  • 3 & 7 are always lucky numbers; 7 x 7 = 49er Gold Miner.

In Number Theory Lucky Numbers are actually defined and can be generated using a sieve somewhat similar to the prime number generating Sieve of Eratosthenes. There is an infinite number of Lucky Numbers, and yes, 3, 7, 9, 21, 49, and 777 all make the list.

Oeis.org’s wiki, Lucky numbers, includes several lucky number lists including the first 33 composite Lucky Numbers thus defined because ALL of their factors are also Lucky Numbers. 777 was the 19th number on that particular list because ALL of its factors, 1, 3, 7, 21, 37, 111, 259, 777 are Lucky Numbers, too!

If that isn’t lucky enough, 777 is a repdigit in three different bases.

  • 3333 BASE 6; note that 3(6^3) + 3(6^2) + 3(6^1) + 3(6^0) = 777
  • 777 BASE 10; note that 7(100) + 7(10) + 7(1) = 777
  • LL BASE 36 (L is 21 base 10); note that 21(36) + 21(1) = 777

Did you notice that lucky numbers 3, 7, and 21 showed  up again? I liked that coincidence so much that I made this graphic:

777 Repdigit

777 is also the sum of three squares four different ways:

  • 26² + 10² + 1² = 777
  • 22² + 17² + 2² = 777
  • 20² + 19² + 4² = 777
  • 20² + 16² + 11² = 777

 

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