A Multiplication Based Logic Puzzle

Posts tagged ‘Pythagorean triples’

785 and Level 4

  • 785 is a composite number.
  • Prime factorization: 785 = 5 x 157
  • 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 785 has exactly 4 factors.
  • Factors of 785: 1, 5, 157, 785
  • Factor pairs: 785 = 1 x 785 or 5 x 157
  • 785 has no square factors that allow its square root to be simplified. √785 ≈ 28.01785.

785-factor-pairs

Now for today’s puzzle:

785-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Here’s a little more about the number 785:

785 is the sum of two squares two different ways:

  • 28² + 1² = 785
  • 23² + 16² = 785

785 is also the sum of three squares three different ways:

  • 26² + 10² + 3² = 785
  • 25² + 12² + 4² = 785
  • 19² + 18² + 10² = 785

Because its prime factorization is 5 × 157 (two numbers that are also the sum of two squares), 785 is the hypotenuse of four Pythagorean triples, two of which are primitive triples:

  • 56-783-785 primitive calculated from 2(28)(1), 28² – 1², 28² + 1²
  • 273-736-785 primitive calculated from 23² – 16², 2(23)(16), 23² + 16²
  • 425-660-785 which is 5 times 85-132-157
  • 471-628-785 which is 157 times 3-4-5

785 is also a palindrome in two different bases:

  • 555 BASE 12; note that 5(144) + 5(12) + 5(1) = 785
  • 101 BASE 28; note that 1(28²) + 0(28) + 1(1) = 785

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785-logic

 

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Can You See How 779’s Factor Pairs Are Hiding in Some Pythagorean Triples?

  • 779 is a composite number.
  • Prime factorization: 779 = 19 x 41
  • 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 779 has exactly 4 factors.
  • Factors of 779: 1, 19, 41, 779
  • Factor pairs: 779 = 1 x 779 or 19 x 41
  • 779 has no square factors that allow its square root to be simplified. √779 ≈ 27.91057.

779-factor-pairs

Those factor pairs are hiding in some Pythagorean triples. Scroll down to read how, but first here’s today’s puzzle:

779 Puzzle

Print the puzzles or type the solution on this excel file: 10-factors-2016

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And now some Pythagorean triple number theory using 779 as an example:

The factors of 779 are very well hidden in five Pythagorean triples that contain the number 779. Here’s how: 779 has two factor pairs:  19 x 41 and 1 x 779. Those factor pairs show up in some way in each of the calculations for these 779 containing Pythagorean triples:

  1. 171-760-779 which is 19 times each number in 9-40-41.
  2. 779-303420-303421, a primitive calculated from 779(1); (779² – 1²)/2; (779² + 1²)/2.
  3. 779-7380-7421 which is 41 times each number in 19-180-181.
  4. 779-15960-15979 which is 19 times each number in 41-840-841.
  5. 660-779-1021, a primitive calculated from (41² – 19²)/2; 19(41); (41² + 19²)/2.

Being able to find whole numbers that satisfy the equation a² + b² = c² is one reason why finding factors of a number is so worth it. ANY factor pair for numbers greater than 2 will produce at least one Pythagorean triple that satisfies a² + b² = c². The more factor pairs a number has, the more Pythagorean triples will exist that contain that number. 779 has only two factor pairs so there are a modest number of 779 containing Pythagorean triples. All of its factors are odd so it was quite easy to find all of the triples. Here’s a brief explanation on how each triple was found:

  1. 799 has one prime factor that has a remainder of 1 when divided by 4. That prime factor, 41, is therefore the hypotenuse of a primitive Pythagorean triple. When the Pythagorean triple is multiplied by the other half of 41’s factor pair, 19, we get a Pythagorean triple in which 779 is the hypotenuse.
  2. Every odd number greater than 1 is the short leg of a primitive Pythagorean triple. To find that primitive for a different odd number, simply substitute the desired odd number in the calculation in place of 779.
  3. Because every odd number greater than 1 is the short leg of a primitive Pythagorean triple, 19(1); (19² – 1²)/2; (19² + 1²)/2 generates the primitive triple (19-180-181). Multiplying each number in that triple by the other half of 19‘s factor pair, 41, produces a triple with 779 as the short leg.
  4. Because every odd number greater than 1 is the short leg of a primitive Pythagorean triple, 41(1); (41² – 1²)/2; (41² + 1²)/2 generates the primitive triple (41-840-841). Multiplying each number in that triple by the other half of 41‘s factor pair, 19, produces a triple with 779 as the short leg.
  5. Since factor pair 19 and 41 have no common prime factors, the formula (41² – 19²)/2; 19(41); (41² + 19²)/2 produces another primitive triple 660-779-1021. If they did have common factors, the factor pair would still produce a triple, but it would not be a primitive one.

Here’s some other interesting facts about the number 779:

779 is the sum of eleven consecutive prime numbers:

47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 = 779.

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

  • 27² + 7² + 1² = 779
  • 27² + 5² + 5² = 779
  • 23² + 15² + 5² = 779
  • 23² + 13² + 9² = 779
  • 21² + 17² + 7² = 779
  • 21² + 13² + 13² = 779

Finally, the table below shows some logical steps that could be used to solve Puzzle #779:

779-logic

 

 

 

745 Pythagorean Triple Puzzle

  • 745 is a composite number.
  • Prime factorization: 745 = 5 x 149
  • 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 745 has exactly 4 factors.
  • Factors of 745: 1, 5, 149, 745
  • Factor pairs: 745 = 1 x 745 or 5 x 149
  • 745 has no square factors that allow its square root to be simplified. √745 ≈ 27.294688.

745-factor-pairs

PUZZLE 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.

Today’s Pythagorean triple puzzle has only 2 more triangles than last week’s puzzle, but it shouldn’t be any more difficult. Please, give it a try!

745 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-01-18

Sorted Triples

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Here are some fun facts about the number 745:

745 can be written as the sum of consecutive numbers three different ways:

  • 372 + 373 = 745; that’s 2 consecutive numbers
  • 147 + 148 + 149 + 150 + 151 = 745; that’s 5 consecutive numbers
  • 70 + 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 = 745; that’s 10 consecutive numbers

745 can also be written as the sum of the thirteen prime numbers from 31 to 83. See if you can name all those primes while you add them up.

745 is the sum of two squares two different ways:

  • 27² + 4² = 745
  • 24² + 13² = 745

Both of 745’s prime factors are hypotenuses of Pythagorean triples, so 745 is the hypotenuse of FOUR Pythagorean triples:

  • 216-713-745; calculated from 2(27)(4), 27² – 4², 27² + 4².
  • 255-700-745
  • 407-624-745; calculated from 2(24)(13), 24² – 13², 24² + 13².
  • 447-596-745

5 is the greatest common factor of one of the non-primitive triples while 149 is the greatest common factor of the other. Which is which?

If you check any of those triples, you will see that 745² is 555025, which is a cool looking number, too.

745 is also the sum of three squares three different ways:

  • 18² + 15² + 14² = 745
  • 22² + 15² + 6² = 745
  • 24² + 12² + 5² = 745

745 is a palindrome in three different bases, two of which are consecutive:

  • 454 BASE 13; note that 4(13²) + 5(13) + 4(1) = 745.
  • 3B3 BASE 14 (B= 11 base 10); note that 3(14²) + 11(14) + 3(1) = 745.
  • 171 BASE 24; note that 1(24²) + 7(24) + 1(1) = 745.

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738 Warning: These 3 right triangles do NOT form one large right triangle

  • 738 is a composite number.
  • Prime factorization: 738 = 2 x 3 x 3 x 41, which can be written 738 = 2 x (3^2) x 41
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 738 has exactly 12 factors.
  • Factors of 738: 1, 2, 3, 6, 9, 18, 41, 82, 123, 246, 369, 738
  • Factor pairs: 738 = 1 x 738, 2 x 369, 3 x 246, 6 x 123, 9 x 82, or 18 x 41
  • Taking the factor pair with the largest square number factor, we get √738 = (√9)(√82) = 2√82 ≈ 27.166155.

738-factor-pairs

The puzzle below 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.

Go ahead, give it a try!

738 Triple Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-11

Sorted Triples

Some time ago I published a rather ambitious Pythagorean triple logic puzzle. I didn’t post the answers but invited anyone who desired to post some or all of the answers and to do so in the comments. As of today, no one has posted any answers. Perhaps that puzzle was too difficult. I decided to post a SIMPLER one today. Just follow the instructions above the puzzle.

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Now for some facts about the number 738:

Because 41 is one of its factors, 738 is the hypotenuse of Pythagorean triple 162-720-738. What is the greatest common factor of the three numbers in the triple? It’s the other number in the same factor pair as 41.

738 can be expressed as the sum of consecutive numbers four different ways:

  • 245 + 246 + 247 = 738; that’s 3 consecutive numbers.
  • 78 + 79 + 80 + 81 + 82 + 83 + 84 + 85 + 86 = 738; that’s 9 consecutive numbers.
  • 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 = 738; that’s 12 consecutive numbers.
  • 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; that’s 36 consecutive numbers.

738 is a palindrome in base 15 and base 16, two consecutive bases:

  • 343 BASE 15; note that 3(225) + 4(15) + 3(1) = 738.
  • 2E2 BASE 16 (E = 14 base 10); note that 2(256) + 14(16) + 2(1) = 738.

From Stetson.edu I learned that 6 + 66 + 666 = 738. Cool! Six 6’s were used!

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725 and Level 2

725 can be expressed as the sum of consecutive numbers five different ways:

  • 362 + 363 = 725; that’s 2 consecutive numbers.
  • 143 + 144 + 145 + 146 + 147 = 725; that’s 5 consecutive numbers.
  • 68 + 69 + 70 + 71 + 72 + 73 + 74 + 75 + 76 + 77 = 725; that’s 10 consecutive numbers.
  • 17 + 18 + 19 + . . . + 29 + . . . + 39 + 40 + 41 = 725; that’s 25 consecutive numbers.
  • 11 + 12 + 13 + . . . + 25 + . . . + 37 + 38 + 39 = 725; that’s 29 consecutive numbers.

725 is also the sum of the eleven prime numbers from 43 to 89.

The factors in one of its factor pairs, 25 x 29, are both 2 numbers away from their average, 27, so 725 is just 4 numbers away from perfect square 27² = 729 . Thus, 25 x 29 =  (27 – 2)(27 + 2) = 27² – 2² = 729 – 4 = 725.

725 is the sum of two squares three different ways:

  • 26² + 7² = 725
  • 25² + 10² = 725
  • 23² + 14² = 725

Because ALL of its prime factors have a remainder of one when divided by four, 725 is the hypotenuse of primitive Pythagorean triples:

  • 364-627-725 which was calculated using 2(26)(7), 26² – 7², 26² + 7²
  • 333-644-725 which was calculated using 23² – 14², 2(23)(14), 23² + 14²

It is also the hypotenuse of FIVE other Pythagorean triples.

  • 85-720-725
  • 120-715-725
  • 203-696-725
  • 435-580-725
  • 500-525-725

725 is a palindrome in two bases:

  • 505 BASE 12; note that 5(144) + 0(12) + 5(1) = 725.
  • PP BASE 28 (P = 25 base 10); note that 25(28) + 25(1) = 725.

725 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-28

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  • 725 is a composite number.
  • Prime factorization: 725 = 5 x 5 x 29, which can be written 725 = (5^2) x 29
  • 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 725 has exactly 6 factors.
  • Factors of 725: 1, 5, 25, 29, 145, 725
  • Factor pairs: 725 = 1 x 725, 5 x 145, or 25 x 29
  • Taking the factor pair with the largest square number factor, we get √725 = (√25)(√29) = 5√29 ≈ 26.925824.

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725 Factors

What Makes 689 Amazing?

689 is an amazing number for several reasons. I decided to make graphics to illustrate many of those ways.

689 is the sum of consecutive prime numbers 227, 229, and 233.

Also 689 is the sum of the primes from 83 to 109. Do you know what those 7 prime numbers are?

Stetson.edu informs us that 689 is the smallest number that can be expressed as the sum of three different square numbers NINE ways. I decided to figure out what those nine ways are and make this first graphic to share with you:

689 Sum of 3 Different Squares

Note: 614 can also be expressed as the sum of 3 squares 9 different ways, but one of those ways is 17² + 17² + 6² = 614, and that duplicates 17² in the same sum.

689 is the same number when it is turned upside down. Numbers with that characteristic are called Strobogrammatic numbers.

689 Rotation

689 BASE 10 isn’t a palindrome, but 373 BASE 14 is; note that 3(196) + 7(14) + 3(1) = 689

Both of 689’s prime factors have a remainder of 1 when divided by 4, so they are hypotenuses of Pythagorean triples. That fact also means 689 can be expressed as the sum of two square numbers TWO different ways, and it makes 689 the hypotenuse of FOUR Pythagorean triples.  Can you tell by looking at the graphic which two are primitive and which two aren’t?

689 Pythagorean triples

689 is the sum of consecutive numbers three different ways:

  • 344 + 345 = 689; that’s 2 consecutive numbers.
  • 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 = 689; that’s 13 consecutive numbers.
  • 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 = 689; that’s 26 consecutive numbers.

Now you have a few reasons why 689 is an amazing number. 13 and 53 were part of some of those reasons so it shouldn’t surprise anyone to see 13 and 53 pop up in its factoring information, too:

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  • 689 is a composite number.
  • Prime factorization: 689 = 13 x 53
  • 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 689 has exactly 4 factors.
  • Factors of 689: 1, 13, 53, 689
  • Factor pairs: 689 = 1 x 689 or 13 x 53
  • 689 has no square factors that allow its square root to be simplified. √689 ≈ 26.248809.

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685 Is the Sum of Two Squares Two Different Ways

Because both of its prime factors are hypotenuses of primitive Pythagorean triples, 685 is the hypotenuse of FOUR Pythagorean triples. Two are primitive; two are not:

  • 37-684-685 which was calculated from 19² – 18², 2(19)(18), 19² + 18²
  • 156-667-685 which was calculated from 2(26)(3), 26² – 3², 26² + 3²
  • 411-548-685 (What factor of 685 is the greatest common factor of those 3 numbers?)
  • 440-525-685 (and what is their greatest common factor?)

As you may have notice from those calculations, 685 is the sum of two squares two different ways:

  • 19² + 18² = 685
  • 26² + 3² = 685

685 is the 19th centered square number because 18 and 19 are consecutive numbers and 19² + 18² = 685. There are 685 small squares of various colors in this graphic.

685 is the 19th Centered Squared Number

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  • 685 is a composite number.
  • Prime factorization: 685 = 5 x 137
  • 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 685 has exactly 4 factors.
  • Factors of 685: 1, 5, 137, 685
  • Factor pairs: 685 = 1 x 685 or 5 x 137
  • 685 has no square factors that allow its square root to be simplified. √685 ≈ 26.172505.

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