## A Multiplication Based Logic Puzzle

### 873 and Level 5

8 + 7 + 3 = 18; 1 + 8 = 9, so 873 can be evenly divided by 9.

27² + 12² = 873 so 873 is the hypotenuse of a Pythagorean triple:

• 585-648-873 which is 9 times (65-72-97), and can be calculated from 27² – 12², 2(27)(12), 27² + 12²

Stetson.edu reminds us that 1! + 2! + 3! + 4! + 5! + 6! = 873.

Print the puzzles or type the solution on this excel file: 12 factors 864-874

• 873 is a composite number.
• Prime factorization: 873 = 3 × 3 × 97, which can be written 873 = 3² × 97
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 873 has exactly 6 factors.
• Factors of 873: 1, 3, 9, 97, 291, 873
• Factor pairs: 873 = 1 × 873, 3 × 291, or 9 × 97
• Taking the factor pair with the largest square number factor, we get √873 = (√9)(√97) = 3√97 ≈ 29.546573

### What Kind of Prime Is 859?

A prime number is a positive number that has exactly two factors, one and itself. (One has only one factor, so it is not a prime number.)

• 859 is the 149th prime number.

A twin prime is a set of two prime numbers in which the second prime number is two more that the first prime number.

• 859 is the second prime number in the 34th twin prime: (857, 859).

A prime triplet is a set of three consecutive prime numbers in which the last number is six more than the first number. Prime triplets always contain a set of twin primes.

• 859 is in the 27th and 28th prime triplets: (853, 857, 859) and (857, 859, 863).

A prime quadruplet is a set of four consecutive prime numbers in which the last number is eight more than the first number. Prime quadruplets always contain TWO sets of overlapping prime triplets.

• Even though prime numbers (853, 857, 859, 863) contain two sets of overlapping prime triplets, they do NOT form a prime quadruplet because the last number is ten more than the first number. Other than (5, 7, 11, 13), all prime quadruplets are prime decades whose last digits are 1, 3, 7, and 9, in THAT order.

There are other prime constellations like prime quintuplets and prime sextuplets, but each of those has to contain a prime quadruplet in it, so 859 isn’t in any of those.

859÷4 = 214 R3. Since that wasn’t R1, we know that 859 is NOT the hypotenuse of ANY Pythagorean triples.

Now you know what kind of prime 859 is.

Here’s today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors-853-863

• 859 is a prime number.
• Prime factorization: 859 is prime.
• The exponent of prime number 859 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 859 has exactly 2 factors.
• Factors of 859: 1, 859
• Factor pairs: 859 = 1 × 859
• 859 has no square factors that allow its square root to be simplified. √859 ≈ 29.3087

How do we know that 859 is a prime number? If 859 were not a prime number, then it would be divisible by at least one prime number less than or equal to √859 ≈ 29.3. Since 859 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 859 is a prime number.

### 851 Give This Apple to Your Teacher This Year

This puzzle looks a little like an apple. It’s a level 5 puzzle so it won’t be that easy. If you can solve the puzzle, give it to your teacher!

Print the puzzles or type the solution on this excel file: 12 factors 843-852

851 is the hypotenuse of a Pythagorean triple:

• 276-805-851 which is 23 times (12-35-37)

851 is a palindrome in three other bases:

• 353 BASE 16, because 3(16²) + 5(16¹) + 3(16º) = 851
• 191 BASE 25, because 1(25²) + 9(25¹) + 1(25º) = 851
• NN BASE 36 (N is 23 base 10) because 23(36¹) + 23(36º) = 23(37) = 851

Here is 851 factoring information:

• 851 is a composite number.
• Prime factorization: 851 = 23 × 37
• The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 851 has exactly 4 factors.
• Factors of 851: 1, 23, 37, 851
• Factor pairs: 851 = 1 × 851 or 23 × 37
• 851 has no square factors that allow its square root to be simplified. √851 ≈ 171904

### 839 and Level 5

839 is the sum of the five prime numbers from 157 to 179:

• 157 + 163 + 167 + 173 + 179 = 839

Print the puzzles or type the solution on this excel file: 10-factors-835-842

• 839 is a prime number.
• Prime factorization: 839 is prime.
• The exponent of prime number 839 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 839 has exactly 2 factors.
• Factors of 839: 1, 839
• Factor pairs: 839 = 1 × 839
• 839 has no square factors that allow its square root to be simplified. √839 ≈ 28.9654967

How do we know that 839 is a prime number? If 839 were not a prime number, then it would be divisible by at least one prime number less than or equal to √839 ≈ 28.97. Since 839 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 839 is a prime number.

### 833 and Level 5

Print the puzzles or type the solution on this excel file: 12 factors 829-834

Wrap Your Mind Around This:  × 17 = 833 = 28² + 7²

833 is the hypotenuse of a Pythagorean triple:

• 392-735-833 calculated from 2(28)(7), 28² – 7², 28² + 7²

833 can be written as the sum of consecutive prime numbers two ways. One of those ways starts with one of its prime factors, 17:

• 833 is the sum of the seventeen prime numbers from 17 to 83
• 833 is also the sum of the eleven prime numbers from 53 to 101

Since 833 has three factor pairs where both factors are odd, it can be written as the difference of two squares three different ways:

• 833 × 1 = 833 means 417² – 416² = 833
• 119 × 7 = 833 means 63² – 56² = 833
• 49 × 17 = 833 means 33² – 16² = 833

The 41st triangular number will be 861. We must use less than 41 consecutive numbers if we want to express 833 as the sum of consecutive numbers. 833 has 3 odd factors (1, 7, 17) less than 41. Thus 833 can be written as the sum of 7 consecutive numbers and as the sum of 17 consecutive numbers. Notice 833’s factor pairs below highlighted in red.

• 833 = 116 + 117 + 118 + 119 + 120 + 121 + 122; that’s 7 consecutive numbers
• 833 = 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57; that’s 17 consecutive numbers

The factor of 833 that is the highest power of 2 is 1 because 2º=1. Each of those odd factors, (1, 7, 17), times 2 × 1 is still less than 40, so 833 can also be written as the sum of 2 consecutive numbers, the sum of 14 consecutive numbers, and the sum of 34 consecutive numbers:

• 833 = 416 + 417; that’s 2 consecutive numbers
• 833 = 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66; that’s 14 consecutive numbers
• 833 = 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 + 40 + 41; that’s 34 consecutive numbers

Here is 833’s factoring information:

• 833 is a composite number.
• Prime factorization: 833 = 7 × 7 × 17, which can be written 833 = 7² × 17
• The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 833 has exactly 6 factors.
• Factors of 833: 1, 7, 17, 49, 119, 833
• Factor pairs: 833 = 1 × 833, 7 × 119, or 17 × 49
• Taking the factor pair with the largest square number factor, we get √833 = (√49)(√17) = 7√17 ≈ 28.861739

### 827 and Level 5

827 is one of the prime numbers in the fourth prime decade, (821, 823, 827, 829).

827 = 103 + 107+ 109 + 113+ 127 + 131 + 137, that’s the sum of 7 consecutive prime numbers.

Here’s today’s puzzle:

Print the puzzles or type the solution on this excel file: 10-factors-822-828

• 827 is a prime number.
• Prime factorization: 827 is prime.
• The exponent of prime number 827 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 827 has exactly 2 factors.
• Factors of 827: 1, 827
• Factor pairs: 827 = 1 × 827
• 827 has no square factors that allow its square root to be simplified. √827 ≈ 28.7576

How do we know that 827 is a prime number? If 827 were not a prime number, then it would be divisible by at least one prime number less than or equal to √827 ≈ 28.8. Since 827 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 827 is a prime number.

### 819 How to Type eˣᵖᵒⁿᵉⁿᵗˢ in WordPress

Exponents, ⁰¹²³⁴⁵⁶⁷⁸⁹, are written to the right of their base numbers and a little higher. They are about half the height and about half the width of the base number, too.

Exponents are important to me. They and other special characters allow me to include factoring information and interesting number facts on every post I write. For example …

• 819 is a composite number.
• Prime factorization: 819 = 3 × 3 × 7 × 13, which can be written 819 = 3² × 7 × 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 × 2 × 2 = 12. Therefore 819 has exactly 12 factors.
• Factors of 819: 1, 3, 7, 9, 13, 21, 39, 63, 91, 117, 273, 819
• Factor pairs: 819 = 1 × 819, 3 × 273, 7 × 117, 9 × 91, 13 × 63, or 21 × 39
• Taking the factor pair with the largest square number factor, we get √819 = (√9)(√91) = 3√91 ≈ 28.618176

1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² + 11² + 12² + 13² = 819, making 819 the 13th square pyramidal number.

315² + 756² = 819² so 819 is the hypotenuse of a Pythagorean triple.

2⁹ + 2⁸ + 2⁵ +  2⁴ + 2¹ + 2⁰  = 819 because 819 is 1100110011 in BASE 2.

I like that pattern of 1’s and 0’s. Here’s a few more of 819’s cool number patterns:

• 3⁶ + 3⁴ + 3² = 819 because 819 is 101010 in BASE 3.
• 3·4⁴ + 3·4² + 3·4⁰ = 819 because 819 is 30303 in BASE 4.
• 3·16² + 3·16¹ + 3·16º = 819 because 819 is 333 in BASE 16.

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Okay. Enough about 819. HOW do we type exponents when we write a blog?

Option #1: WordPress gives us some special characters in the editor. I’ve put red boxes around the exponents so you can find them faster:

As you can see, the WordPress’s editor only offers us º ¹ ² ³ ª as exponents, and they MIGHT fill all your needs. (Who am I kidding?) You can get to any of the symbols shown above by clicking on the Ω symbol in YOUR WordPress editor. I’ve put a red box around the Ω special character symbol in the PICTURE of the editor below.

Those symbols are good if you’re writing x³ or even 8¹º³². But what if you want to write an expression with a 4, 5, 6, 7, 8, or 9 as part of the exponent? Do you really have to settle for (2^7)×(3^5) when you really want to type 2⁷×3⁵? That carrot ^ symbol can look needlessly intimidating to people even if they are familiar with exponents.

So how do we type all those other exponents in WordPress? That is something I have been frustrated about and have googled about many times. I’ve read about and tried a couple more options: Superscripts and LaTeX.

Option #2 Superscripts: When I followed the superscripts’ directions for WordPress, and typed e<sup>xponents</sup> in the text editor as instructed, it made beautiful eˣᵖᵒⁿᵉⁿᵗˢ in the visual editor, but look at all these exponents marked in red, they fell down when I published this post. That is not acceptable. Other people may be able to get those superscripts to stay up, but I have tried repeatedly without success for over a year. On a related note: While writing this post I learned something useful about Microsoft Word. If you push down the Shift, Ctrl, and = keys at the same time, you can type in superscript in a Microsoft Word document. (You press the same keys to get out of superscript mode). Unfortunately, if you copy and paste that superscript writing into WordPress, it will look like superscripts in the visual editor, but not in your published work.

Option #3. LaTeX can be a great looking option. Still, when the exponents from the WordPress editor are typed alongside those in LaTeX, they can look a little wobbly: 2³ + $2^4$ + $2^5$ + 7² + $3^4$. In addition, LaTeX looked like LaTeX notation instead of exponents when I tried to use it in the title of this post.  The biggest drawback: LaTeX looks good when it’s published, but it is practically unreadable when it’s being typed. For example, without spaces, you must type [ latex ]2^4[ /latex ], just to get 2⁴. (If I took out the spaces it would read 2⁴ instead of showing you what LaTeX notation looks like.)

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This week I found a good 4th option: Microsoft Word has quite a few exponents, and WordPress liked them!

I’ve gathered the superscripts of the English alphabet and numbers from Microsoft Word in one place and included them here for the convenience of all other bloggers, making this post a great 5th option. Copy what you need from here, or copy and paste the whole list into a handy document of your own. True, not every letter of the English alphabet is available as an exponent in Word, but most of them are. This is the method I used to include eˣᵖᵒⁿᵉⁿᵗˢ in the title of this post.

x⁰¹²³⁴⁵⁶⁷⁸⁹ᴬᵃᵅᴮᵇᶜᴰᵈᴱᵉᶠᴳᵍᴴʰᴵⁱᴶʲᴷᵏᴸᴹᵐᴺⁿᴼᵒᴾᵖᴿʳˢᵀᵗᵁᵘⱽᵛᵂʷˣʸᶻ ⁺ ⁻ ⁼ ⁽  ⁾ Those superscripts or exponents stay up! And…these subscripts stay down! ₉₈₇₆₅₄₃₂₁₀ ₊ ₋ ₌ ₍ ₎ aₐ eₑ  jⱼ oₒ xₓ. Curiously,  hₕ kₖ lₗ mₘ nₙ pₚ sₛ tₜ seem to stay down on home computers but disappear on smart phones so you might not want to use them.

Subscripts are often used in notation for Permutations like ₆P₃ or Combinations like ₆C₃. (Those links will take you to some useful online calculators.)  Subscripts used with “⅟ ” can write infinitely many unit fractions like ⅟₃₂₁. Subscripts can also be used to write the base of logarithms such as log₂4=2.

Here’s a bonus, the Greek letters: Some of the Greek letters have superscripts and/or subscripts next to them, while others do not. For some reason unknown to me, Microsoft Word didn’t give π either one. (I could not have written this part of the post without zooming to 175% first. You might want to do that before using any of these, too.):

Ααᵅ, Ββᵝᵦ, Γ⸀γᵞᵧ, Δδᵟ, Εεᵋ, Ζᶻζ, Ηη, Θᶱθᶿ, Ιᶦιᶥ, Κκ, Λᶺλ, Μμ, Νᶰν, Ξξ, Οο, Ππ, Ρρ, Σσ, Ττ, Υυᶹ, Φᶲφᵠᵩ, Χχᵡᵪ, Ψψ, Ωω

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Those exponents from Microsoft Word will allow you to write important identities like the following without using awkward LaTeX notation:

• sin t = (eⁱᵗ – e־ⁱᵗ)/2i
• cos t = (eⁱᵗ + e־ⁱᵗ)/2

Back to the 4th option, Microsoft Word does include some other incomplete alphabets from other languages that are not included in my lists above. Here’s what you’ll need to do to get subscripts or superscripts from Microsoft Word:

1. In Microsoft Word click on “insert”,
2. click on “symbol”,
3. click on “symbol” (NOT “equation” because WordPress won’t copy anything you type there),
4. click on “more symbols”.
5. Next LOOK for the desired superscript or subscript on the chart. You may have to look for a while. Some of the them are listed together, while others seem to be randomly placed by themselves. For the alphabet, only use a letter that is in the top CENTER of its box. If you use a letter that is in the top LEFT of its box, you might end up typing something like 3 ͩͪ  or 7ͪͫ.
6. Type your expression in Word, then copy and paste it onto your blog.

So now you have been saved countless hours of frustration trying to type a few simple exponents or subscripts. Perhaps, now you can chance getting frustrated trying to solve this Level 5 puzzle?!

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