A Multiplication Based Logic Puzzle

Posts tagged ‘Prime factorization’

Numbers up to 820 with Exactly 12 Factors

Let’s begin with today’s puzzle. Afterwards I’ll tell you a little about the number 820 and why I decided to make a list of all the numbers up to 820 with exactly 12 factors.

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

 

Now here’s some information about the number 820:

  • 820 is a composite number.
  • Prime factorization: 820 = 2 × 2 × 5 × 41, which can be written 820 = 2² × 5 × 41
  • 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 820 has exactly 12 factors.
  • Factors of 820: 1, 2, 4, 5, 10, 20, 41, 82, 164, 205, 410, 820
  • Factor pairs: 820 = 1 × 820, 2 × 410, 4 × 205, 5 × 164, 10 × 82, or 20 × 41
  • Taking the factor pair with the largest square number factor, we get √820 = (√4)(√205) = 2√205 ≈ 28.635642

820 is the sum of two squares two ways:

  • 28² + 6² = 820
  • 26² + 12² = 820

Because 5 and 41 are two of its prime factors, 820 is the hypotenuse of FOUR Pythagorean triples:

  • 180-800-820 which is 20 times 9-40-41
  • 336-748-820 which is 4 times 84-187-205, but it could also be calculated from 2(28)(6), 28² – 6², 28² + 6²
  • 492-656-820 which is 164 times 3-4-5
  • 532-624-820 which is 4 times 133-156-205, but it could also be calculated from 26² – 12², 2(26)(12),26² + 12²

Since 820 = (40×41)/2, we know that 820 is the 40th triangular number, thus

  • 1 + 2 + 3 + . . . + 38 + 39 + 40 = 820

820 is also a palindrome in four other bases:

  • 1010101 BASE 3 because 3⁶+3⁴+3²+3⁰=820
  • 1111 BASE 9 because 9³+9²+9¹+9⁰=820
  • 868 BASE 11 because 8(11²)+6(11¹)+8(11⁰)=820
  • 1I1 BASE 21 (I is 18 base 10) because 21²+18(21¹)+21⁰=820

Below is a chart of the numbers up to 820 with exactly 12 factors. Notice that two sets of consecutive numbers, (735, 736) and (819, 820), are on the list. Look at their prime factorizations:

  • 735=3×5×7², 736=2⁵×23
  • 819=3²×7×13, 820=2²×5×41

Those prime factorizations mean that while 735 and 736 are the smallest consecutive numbers with exactly 12 factors, 819 and 820 are the smallest consecutive numbers whose prime factorizations consist of one prime number squared and exactly two other primes. Thanks to Stetson.edu for alerting me to that fact. Here’s something interesting about the chart: of the 77 numbers listed, only six are odd numbers.

So, how did I know what numbers to put on the list?

In order to determine how many numbers up to 820 have exactly 12 factors, we must first factor 12. We know that 12=12, 6×2, 4×3, and 3×2×2.

Next we subtract 1 from each of those factors to determine the exponents we need to use:

12 gives us 12-1=11. For prime number a, when is a¹¹ not larger than 820? Never, because 2¹¹>820.

6×2 gives us 6-1=5 and 2-1=1. For prime numbers a and b, with a≠b, when is ab¹ less than or equal to 820? These nine times:

  • 2×3=96, 2×5=160, 2×7=224, 2×11=352,
  • 2×13=416, 2×17=544, 2×19=608, 2×23=736
  • 3×2=486

4×3 gives us 4-1=3 and 3-1=2. For prime numbers a and b, with a≠b, when is a³b² not larger than 820? These six times:

  • 2³×3²=72, 2³×5²=200, 2³×7²=392
  • 3³×2²=108, 3³×5²=675
  • 5³×2²=500

3×2×2 gives us 3-1=2, 2-1=1, and 2-1=1. For distinct prime numbers a, b, and c, when is a²bc not larger than 820? 52 times. Here’s the breakdown: It happens 35 times when 2²=4 is the square number:

And it happens another 27 times when a prime number other than 2 is squared:

 

That’s a lot of numbers with exactly 12 factors! After I sorted all the numbers that I found in numerical order, I was able to make that chart of numbers up to 820 with exactly 12 factors, and yes 819 and 820 are the smallest two consecutive numbers whose prime factorization consists exactly of one prime number squared and two other prime numbers.

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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

 

818 How Many Steps Do You Take Each Day?

My brother, Doug, recently visited me. He told me about his goal to get 11,000 steps every day. The American Heart Association recommends 10,000 steps a day.  Is it worth trying to get a thousand steps more than the recommended number?

My brother shared the cool mathematics of an 11,000 daily step goal with me, and now I want to share it with you:

Now I think 11,000 steps a day is a very worthy goal! It helps me see the big picture of 1,000,000 steps each quarter and 4,000,000 steps each year and that will help me be more likely to meet the 11,000 step goal EVERY day.

I wear a Fitbit to keep track of my steps everyday. A fellow blogger recently wrote a fun poem about wearing a Fitbit to keep track of steps, and it made me smile.

It will take you far fewer than 11,000 steps to complete this multiplication table puzzle. It isn’t the most difficult puzzle I make, but it can still be a challenge:

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

Now here is a little bit about the number 818:

  • 818 is a composite number.
  • Prime factorization: 818 = 2 x 409
  • 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 818 has exactly 4 factors.
  • Factors of 818: 1, 2, 409, 818
  • Factor pairs: 818 = 1 x 818 or 2 x 409
  • 818 has no square factors that allow its square root to be simplified. √818 ≈ 600699

818 looks the same right side up or upside down so we call it is a strobogrammatic number.

23² +  17² = 818

Finally, 818 can be found in these two Pythagorean triple equations:

  • 240² + 782² = 818²
  • 818² + 167280² = 167282²

 

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:

815 and Level 1

Since multi-digit 815 ends with 5, it is a composite number, and it is also the hypotenuse of a Pythagorean triple:

  • 489-652-815 which is 163 times 3-4-5.

Can you write the numbers 1 – 12 in both the first column and the top row so that this puzzle functions like a multiplication table?

 

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

  • 815 is a composite number.
  • Prime factorization: 815 = 5 x 163
  • 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 815 has exactly 4 factors.
  • Factors of 815: 1, 5, 163, 815
  • Factor pairs: 815 = 1 x 815 or 5 x 163
  • 815 has no square factors that allow its square root to be simplified. √815 ≈ 5482048.

813 My Newest Grandchild

My NEWEST grandchild was adopted a couple of months ago in China. My daughter-in-law blogged about picking up their daughter and returning to her orphanage to say good-bye before they left China. The details given are very moving. In spite of the traumatic start, this little girl and her family have grown to love each other very much.

Here she is sitting with my husband, me, and her big sister. My husband and I are pretty new to her so she’s probably thinking in Mandarin, “Who are these people?” On the other hand, we are delighted to be a part of her life now.

Here is today’s puzzle:

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

Here’s a little about the number 813:

  • 813 is a composite number.
  • Prime factorization: 813 = 3 x 271
  • 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 813 has exactly 4 factors.
  • Factors of 813: 1, 3, 271, 813
  • Factor pairs: 813 = 1 x 813 or 3 x 271
  • 813 has no square factors that allow its square root to be simplified. √813 ≈ 51315.

813 is repdigit 111 in BASE 28 because 28² + 28¹ + 28º = 813

Stetson.edu and my calculator informed me that 813^e = 81,366,615.06223032 . . .

 

812 How Many Triangles in All?

Here’s a puzzle for you. How many total triangles are there in the figure below?

Okay, if you guessed 812 because this is my 812th post, you would be right. But what if you were asked that question in some real world situation where accessing the internet to get the answer isn’t permitted. How would you know the answer then?

True, you might have memorized the formula I mentioned in 658-How Many Triangles Point Up? How Many Triangles Point Down? How Many Triangles in All?:

  • The total number of triangles = ⌊n(n+2)(2n+1)/8⌋ where the brackets mean round decimals DOWN to the closest integer. 
  • Here n = 14, so the number of triangles is 14×16×29/8 = 812. Rounding down wasn’t necessary since the product of two consecutive even numbers is always divisible by 8.

Still, you probably wouldn’t remember that formula unless you had seen it VERY recently or you have a photographic memory.

You could actually COUNT all the triangles. In the post about 658 total triangles, I noted that the 13 rows of small triangles formed a total of 169 of the smallest triangles, but I suggested that it would be easier to ignore that nice square number and instead count the number of triangles pointing UP separately from the number pointing DOWN. You will add up a lot of triangular numbers as you sum up the number of them pointing up and again as you sum up the number pointing down.

Making a chart of the number of triangles pointing UP would be easy. It’s just a list of triangular numbers in order. However, the chart for the ones pointing down might be confusing because you don’t use all of the triangular numbers, and the ones you use will be different for an even number of rows than for an odd number of rows. For example,

  • the pointing DOWN portion of the chart for 14 rows of triangles below uses these seven triangular numbers: 1, 6, 15, 28, 45, 66, and 91,
  • while the pointing DOWN chart for 13 rows of triangles uses six different triangular numbers: 3, 10, 21, 36, 55, and 78.

Here is a chart listing the number of triangles of any size that are contained in a triangular figure made with 14 rows of small triangles. Interesting note: Because 14 + 2 = 16, a multiple of 8, the total number of triangles in this case will be divisible by 14, the number of line segments on each side.

Making such a chart works. However, remembering  which triangular numbers to use and how many you should use, especially when counting the odd number of rows of triangles pointing DOWN, might be difficult.

Today I was thinking about triangular numbers and their relationship to square numbers:

  • The sum of two consecutive triangular numbers always makes a square number. On the chart above, I’ve paired up certain consecutive triangular numbers by coloring them the same color. The sizes of the triangles being paired together by like colors are not the same size (with one exception), but their sums are nevertheless square numbers every time.
  • Thus, we also can come up with a single list of numbers to add to arrive at the total number of triangles as illustrated below. This has made me changed my mind about using those square numbers to help count! Notice how every other number on each list below, including the last number to be added, is a square number.

Of the methods discussed in this post, this one that includes square numbers might be the easiest one to remember. Here are the steps I used to find the total number of triangles:

  • Count the rows and determine if that number is even or odd.
  • Write the triangular numbers in order until the amount of numbers written equals the number of rows.
  • If the number of rows is even, replace the 2nd, 4th, 6th, etc. entries with corresponding square numbers: 4, 16, 36, etc.
  • If the number of rows is odd, replace the 1st, 3rd, 5th, etc entries with corresponding square numbers: 1, 9, 25, etc.
  • Add all the numbers remaining in the list. The sum will be the total number of triangles for that many rows of small triangles.

Okay, that puzzle was rather difficult and took a while to explain. You may find this Level 4 puzzle to be easier:

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

Here’s a little more about the number 812:

  • 812 is a composite number.
  • Prime factorization: 812 = 2 x 2 x 7 x 29, which can be written 812 = (2^2) x 7 x 29
  • 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 812 has exactly 12 factors.
  • Factors of 812: 1, 2, 4, 7, 14, 28, 29, 58, 116, 203, 406, 812
  • Factor pairs: 812 = 1 x 812, 2 x 406, 4 x 203, 7 x 116, 14 x 58, or 28 x 29
  • Taking the factor pair with the largest square number factor, we get √812 = (√4)(√203) = 2√203 ≈ 28.4956 (That was three multiples of 7)

812 = 28 × 29, which means it is the sum of the first 28 even numbers.

  • Thus, 2 + 4 + 6 + 8 + . . .  + 52 + 54 + 56 = 812

It also means that we are halfway between 28² and 29², or halfway between 784 and 841. The average of those two numbers is 812.5.

AND it means that 28² + 29² – 1 = 2(812) = 2(28 × 29)

Since 29 is one of its factors, 812 is also the hypotenuse of a Pythagorean triple:

  • 560-588-812 which is 28 times 20-21-29.

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