Pythagorean triples

520 and Level 5

/ ivasallay / Leave a comment

520 is the hypotenuse of four Pythagorean triples. Can you find the greatest common factors of each of these triples:

  • 128-504-520
  • 200-480-520
  • 264-448-520
  • 312-416-520

520 = (23^2) – (3^2).

520 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-08

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  • 520 is a composite number.
  • Prime factorization: 520 = 2 x 2 x 2 x 5 x 13, which can be written 520 = (2^3) x 5 x 13
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 520 has exactly 16 factors.
  • Factors of 520: 1, 2, 4, 5, 8, 10, 13, 20, 26, 40, 52, 65, 104, 130, 260, 520
  • Factor pairs: 520 = 1 x 520, 2 x 260, 4 x 130, 5 x 104, 8 x 65, 10 x 52, 13 x 40, or 20 x 26
  • Taking the factor pair with the largest square number factor, we get √520 = (√4)(√130) = 2√130 ≈ 22.8035085

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485 and Level 5

/ ivasallay / 2 Comments

485 is the hypotenuse of four Pythagorean triples. Which ones are primitive and which ones aren’t?

  • 44-483-485
  • 93-476-485
  • 291-388-485
  • 325-360-485

485 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-05-04

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  • 485 is a composite number.
  • Prime factorization: 485 = 5 x 97
  • 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 485 has exactly 4 factors.
  • Factors of 485: 1, 5, 97, 485
  • Factor pairs: 485 = 1 x 485 or 5 x 97
  • 485 has no square factors that allow its square root to be simplified. √485 ≈ 22.0227155

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485 Logic

480 The Very Inspiring Blogger Award

/ ivasallay / 2 Comments

I Was Nominated for the Very Inspiring Blogger Award!

Nikita Rath recently nominated me for the Very Inspiring Blogger Award. I have really enjoyed reading her blog about her travel adventures across the world. She was born in India and has been to several other very interesting places. I especially loved reading about her trip to Budapest. I was also quite thrilled to read that her favorite school subject was mathematics.

Who I Nominate for the Very Inspiring Blogger Award:

  1. Even though Homeschoolpdx isn’t as comfortable with mathematics as she’d like to be, she has found some very good ways to teach her young children mathematical concepts using storybooks, games, cooking, gardening, and toys! Her children are already very good at math, and I am confident that they will continue to be.
  2. Life Through a Mathematician’s Eyes posted favorites for the month for April: favorite mathematical quote, favorite art and maths inspiration, favorite number, favorite mathematician, and favorite blog/pages/people. This blog is well written and quite pleasing to the eye and was the host of the 121st edition of Carnival of Mathematics.
  3. Lisa M. Peek wrote a post with a very intriguing title: 3-reasons-that-blog-posts-with-numbers-are-popular. She has noticed that blog posts with lists often go viral on facebook. She has given some thought about why that happens and gives some compelling reasons.
  4. Mopdog did the A through Z challenge on 26 Ways to Die in Medieval Hungary. I loved reading every single post. These are stories familiar to every Hungarian but are generally unknown to the rest of the world until these posts were written. A: by Adultery is the best place to start.
  5. Remember how fun it was to use your thumb and a flip book to make a cartoon character dance? Paula Beardell Krieg has spent months planning and preparing four wonderful flip books that teach and reach students who are learning to graph linear equations. She has even prepared pdf’s of the pages that can be downloaded and assembled. Complete instructions are given in the-animated-equation-book.

A Factor Tree for 480:

Although I could make a forest of the many different 480 factors trees, I will only include one of the MANY possible trees here:

Factors of 480:

  • 480 is a composite number.
  • Prime factorization: 480 = 2 x 2 x 2 x 2 x 2 x 3 x 5, which can be written 480 = (2^5) x 3 x 5
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 x 2 x 2 = 24. Therefore 480 has exactly 24 factors.
  • Factors of 480: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 32, 40, 48, 60, 80, 96, 120, 160, 240, 480
  • Factor pairs: 480 = 1 x 480, 2 x 240, 3 x 160, 4 x 120, 5 x 96, 6 x 80, 8 x 60, 10 x 48, 12 x 40, 15 x 32, 16 x 30, or 20 x 24
  • Taking the factor pair with the largest square number factor, we get √480 = (√16)(√30) = 4√30 ≈ 21.9089

Sum Difference Puzzles:

30 has four factor pairs. One of those pairs adds up to 13, and  another one subtracts to 13. Put the factors in the appropriate boxes in the first puzzle.

480 has twelve factor pairs. One of the factor pairs adds up to 52, and a different one subtracts to 52. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

More about the Number 480:

480 is the sum of consecutive primes two different ways:
239 + 241 = 480; those 2 consecutive primes also happen to be twin primes.
109 + 113 + 127 + 131 = 480; that’s 4 consecutive primes.

No counting number less than 480 has more factors than 480 has, but 360 and 420 each have just as many.

Since 24 x 20 is one of its factor pairs, and the difference between those two numbers is 4, the next perfect square is only 4 numbers away. The next perfect square is 484 which is 22 x 22. (This fact is a natural consequence of the fact that 2 + 2 = 4 and 2 x 2 = 4. Only numbers that are 4 less than a perfect square can claim a factor pair with a difference of 4.)

480 is the hypotenuse of the Pythagorean triple 288-384-480.

480 is the longer leg of the primitive Pythagorean triple 31-480-481. Since 480 has so many factors that are divisible by 4, it is in too many other Pythagorean triples to list here.

One of my readers gave another very interesting fact about the number 480 in the comments. Check it out!

 

457 A Pythagorean Triple Logic Puzzle

/ ivasallay / 7 Comments

457 = 4² + 21², and it is the hypotenuse of the primitive Pythagorean triple 168-425-457. Also, 457 is the sum of some consecutive prime numbers. One of my readers posted those primes in the comments.

A long time ago I decided that Pythagorean triples could make a great logic puzzle, so I created one. You can see it directly underneath the following directions:

This puzzle is NOT drawn to scale. Angles that are marked as right angles are 90 degrees, but any angle that looks like a 45 degree angle, isn’t 45 degrees. Lines that look parallel are NOT parallel. Shorter looking line segments may actually be longer than longer looking line segments. Most rules of geometry do not apply here: in fact non-adjacent triangles in the drawing might actually overlap.

No geometry is needed to solve this puzzle. All that is needed is the table of Pythagorean triples under the puzzle. The puzzle only uses triples in which each leg and each hypotenuse is less than 100 units long. The puzzle has only one solution.

If any of these directions are not clear, let me know in the comments. I will NOT be publishing the solution to this puzzle, but I will allow anyone who desires to put any or all of the missing values in the comments. Also, the comments will help me determine if I should publish another puzzle like this one.

Good Luck!

457 Puzzle

Sorted Triples

Print the puzzles or type the solution on this excel file:  10 Factors 2015-04-13

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  • 457 is a prime number.
  • Prime factorization: 457 is prime.
  • The exponent of prime number 457 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 457 has exactly 2 factors.
  • Factors of 457: 1, 457
  • Factor pairs: 457 = 1 x 457
  • 457 has no square factors that allow its square root to be simplified. √457 ≈ 21.3776

How do we know that 457 is a prime number? If 457 were not a prime number, then it would be divisible by at least one prime number less than or equal to √457 ≈ 21.3776. Since 457 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 457 is a prime number.

445 and Level 4

/ ivasallay / 11 Comments

Supposedly there is something interesting about every number. What is interesting about the number 445? When you write it in base 9, the digits are reversed. How do you convert 445 from base 10 to base 9?

Solution: The powers of 9 less than 445 in descending order are 81, 9, and 1. We first divide 445 by 81, next we divide the remainder by 9, and lastly, we divide that remainder by 1 as illustrated below.

445 base 9

Thus 445 (base 10) = 544 (base 9).

445 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-30

  • 445 is a composite number.
  • Prime factorization: 445 = 5 x 89
  • 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 445 has exactly 4 factors.
  • Factors of 445: 1, 5, 89, 445
  • Factor pairs: 445 = 1 x 445 or 5 x 89
  • 445 has no square factors that allow its square root to be simplified. √445 ≈ 21.0950

445 is the hypotenuse of four Pythagorean triples:

  • [267-356-445] which is [3-4-5] times 89
  • [195-400-445] which is [39-80-89] times 5
  • Primitive [84-437-445] (Note: 84 = 2(2 x 21) and 437 = 21^2 – 2^2, while 445 = 21^2 + 2^2)
  • and Primitive [203-396-445] (Note: 396 = 2(18 x 11) and 203 = 18^2 – 11^2, while 445 = 18^2 + 11^2)

445 Logic

 

425 and Level 6

/ ivasallay / Leave a comment

425 ends in 25 so it can be divided evenly by 25. If I had $4.25 all in quarters. How many quarters would I have? That’s the problem that I think of when I divide by 25. All of the factors of 425 are listed below the puzzle.

For some reason unknown to me, here in the United States, dates are ordered by month, date, and year. This rather illogical way of ordering allows us to say that today is 3-14-15, which are the first five digits of pi.  It could also be said that 3-14-15 at 9:26:53 gives the first ten digits of pi.

Logical or not, it is fun to declare today as Pi Day. Today’s puzzle celebrates those first five digits:

425 Puzzle

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-09

  • 425 is a composite number.
  • Prime factorization: 425 = 5 x 5 x 17, which can be written 425 = (5^2) x 17
  • 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 425 has exactly 6 factors.
  • Factors of 425: 1, 5, 17, 25, 85, 425
  • Factor pairs: 425 = 1 x 425, 5 x 85, or 17 x 25
  • Taking the factor pair with the largest square number factor, we get √425 = (√25)(√17) = 5√17 ≈ 20.6155

425 and all of it factors (except 1) are hypotenuses of primitive Pythagorean triples, so 425 is the hypotenuse of several triples:

  • [87-416-425] and
  • [297-304-425] are primitives
  • [65-420-425] is [13-84-85] times 5
  • [119-408-425] is [7-24-25] times 17
  • [180-385-425] is [36-77-85] times 5
  • [200-375-425] is [8-15-17] times 25
  • [255-340-425] is [3-4-5] times 85

425 Logic

416 and Level 5

/ ivasallay / 2 Comments

416 is part of many Pythagorean triples. The smallest one is [160-384-416]. The largest one is the only primitive of the lot: [416-43,263-43,265]. Even though I find these triples incredibly easy to find, you can rest assured that you will probably go your entire life without ever having to know either of them!

416 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-02

  • 416 is a composite number.
  • Prime factorization: 416 = 2 x 2 x 2 x 2 x 2 x 13, which can be written 416 = (2^5) x 13
  • The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 x 2 = 12. Therefore 416 has exactly 12 factors.
  • Factors of 416: 1, 2, 4, 8, 13, 16, 26, 32, 52, 104, 208, 416
  • Factor pairs: 416 = 1 x 416, 2 x 208, 4 x 104, 8 x 52, 13 x 32, or 16 x 26
  • Taking the factor pair with the largest square number factor, we get √416 = (√16)(√26) = 4√26 ≈ 20.3961

 416 Logic

410 Is The Sum of Two Squares Two Different Ways

/ ivasallay / 5 Comments

410 = 59 + 61 + 67 + 71 + 73 + 79, six consecutive prime numbers.

410 = 199 + 211, two consecutive prime numbers.

  • 410 is a composite number.
  • Prime factorization: 410 = 2 x 5 x 41
  • 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 410 has exactly 8 factors.
  • Factors of 410: 1, 2, 5, 10, 41, 82, 205, 410
  • Factor pairs: 410 = 1 x 410, 2 x 205, 5 x 82, or 10 x 41
  • 410 has no square factors that allow its square root to be simplified. √410 ≈ 20.2485

Those factors hardly seem like anything to write home about, but wait….

410 equals

How do I know all this? I applied something I read on Dr. David Mitchell’s blog, Lattice Labyrinths. He recently wrote a fascinating post about Tessellations and Pythagorean Triples. You have to read the article or at least look at his tessellations!

I didn’t know multiplying tessellating lattice labyrinths had anything to do with Pythagorean triples, but they do! (Actually I didn’t know anything about tessellating lattice labyrinths until I read the post.)

About the fourth or fifth paragraph of Dr. Mitchell’s post, he states something I didn’t know before: If you take one number that is the sum of two squares and find another number that is also the sum of two squares, and then if you multiply those two numbers together, that product will also be the sums of two squares! Since I knew both 10 and 41 could be written as the sum of two squares, I had to see if 410 could be also. I was doubly surprised and certainly not the least bit disappointed. I don’t know if it’s unusual that most of 410’s factors can also be written as the sum of two squares.

Because 410 and so many of its factors have that property, it is the hypotenuse of four non-primitive Pythagorean triples:

  • [266-312-410] is [133 – 156 – 205] times 2.
  • [246-328-410] is [3-4-5] times 82.
  • [168-374-410] is [84-187-205] times 2.
  • [90-400-410] is [9-40-41] times 10.

It is also the short leg of four other triples:

  • [410-984-1066] is [5-12-13] times 82.
  • [410-1656-1706] is [205-828-853] times 2.
  • [410-8400-8410] is [41-840-841] times 10.
  • [410-42024-42026] is [205-21012-21013] times 2.