A Multiplication Based Logic Puzzle

Posts tagged ‘Pythagorean triples’

865 and Level 1

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

865 is the sum of two squares two different ways:

  • 28² + 9² = 865
  • 24² + 17² = 865

865 is the hypotenuse of four Pythagorean triples, two of which are primitives:

  • 260-825-865, which is 5 times (52-165-173)
  • 287-816-865, which is 24² – 17², 2(24)(17), 24² + 17²
  • 504-703-865 which is 2(28)(9), 28² – 9², 28² – 9²
  • 519-692-865, which is (3-4-5) times 173

You could see 865’s factors in two of those Pythagorean triples, and here they are again:

  • 865 is a composite number.
  • Prime factorization: 865 = 5 × 173
  • 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 865 has exactly 4 factors.
  • Factors of 865: 1, 5, 173, 865
  • Factor pairs: 865 = 1 × 865 or 5 × 173
  • 865 has no square factors that allow its square root to be simplified. √865 ≈ 29.41088
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How Often is 850 the Hypotenuse of a Pythagorean Triple?

We can tell if a number is the hypotenuse of a Pythagorean triple by looking at its prime factorization.

  • If NONE of its prime factors leave a remainder of 1 when divided by 4, then it will NOT be the hypotenuse of Pythagorean triple.
  • If at least one of its prime factors leave a remainder of 1 when divided by 4, then it WILL be the hypotenuse of Pythagorean triple.
  • If ALL of its prime factors leave a remainder of 1 when divided by 4, then it will also be the hypotenuse of at least one PRIMITIVE Pythagorean triple.

850 = 2 × 5² × 17¹. Its factor, 2, prevents 850 from being the hypotenuse of a primitive Pythagorean triple, but 5² × 17¹ will actually make it the hypotenuse of SEVEN Pythagorean triples. Some of those we can find by looking at the ways we can make 850 from the sum of two squares:

29² + 3² = 850, 27² + 11² = 850, and 25² + 15² = 850

  • 29² + 3² gives us 174-832-850, calculated from 2(29)(3), 29² – 3², 29² + 3², and is 2 times (87-416-425)
  • 27² + 11² gives us 594-608-850, calculated from 27² – 11², 2(27)(11), 27² + 11², and is 2 times (297-304-425)
  • 25² + 15² gives us 400-750-850 calculated from 2(25)(15), 25² – 15², 25² + 15², and is (8-15-17) times 50.

Let’s look a little closer at 850’s factoring information:

  • 850 is a composite number.
  • Prime factorization: 850 = 2 × 5 × 5 × 17, which can be written 850 = 2 × 5² × 17
  • 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 × 3 × 2 = 12. Therefore 850 has exactly 12 factors.
  • Factors of 850: 1, 2, 5, 10, 17, 25, 34, 50, 85, 170, 425, 850
  • Factor pairs: 850 = 1 × 850, 2 × 425, 5 × 170, 10 × 85, 17 × 50, or 25 × 34
  • Taking the factor pair with the largest square number factor, we get √850 = (√25)(√34) = 5√34 ≈ 29.154759.

Here are the SEVEN ways 850 is the hypotenuse of a Pythagorean triple with five of 850’s factor pairs in bold print:

  • 174-832-850 which is 2 times (87-416-425)
  • 594-608-850 which is 2 times (297-304-425)
  • 510-680-850 which is (3-4-5) times 170.
  • 130-840-850 which is 10 times (13-84-85).
  • 360-770-850 which is 10 times (36-77-85).
  • 400-750-850 which is (8-15-17) times 50
  • 238-816-850 which is(7-24-25) times 34

When I wrote about 845, I said I would explore a conjecture a little more:

My conjecture: If prime numbers x and y are Pythagorean triple hypotenuses, and A and B are integers with B ≥ A and A ≥ 1, then xᴬ × y will have two primitive triples. The total number of triples xᴬ × yᴮ will have will be A + B + 2Bᴬ

So…how many Pythagorean triples does 2 × 5³ × 17¹ = 4250 have? It will have the same number as 5³ × 17¹ = 2125.

From the conjecture I figure that 2125 and 4250 will each have 1 + 3 + 2(3¹) = 10 total triples. Let’s see if I’m right …

Besides 1 and itself, the factors of 2125 are 5, 17, 25, 85, 125, and 425, all Pythagorean triple hypotenuses. Each of their respective primitive Pythagorean triples has a multiple with 4250 as the hypotenuse:

  1. 850 times 5’s primitive
  2. 250 times 17’s primitive
  3. 175 times 25’s primitive
  4. 34 times 125’s primitive
  5. 50 times 85’s two primitives
  6. 10 times 425’s two primitives

That’s a total of 8 Pythagorean triples from that list. We will also have triples that are 2 times 2125’s primitives. We can find those triples by looking at the sums of two squares that equal 4250.

  1. 65² + 5² = 4250; but 5 is a factor of both 65 and 5, so this will produce a duplicate of one of the triples already given.
  2. 61² + 23² = 4250; gives us 2806-3192-4250, calculated from 2(61)(23), 61² – 23², 61² + 23²
  3. 55² + 35² = 4250; but 5 is a factor of both 55 and 35, so this will produce a duplicate of one of the triples already given.
  4. 49² + 43² = 4250; gives us 552-4214-4250, calculated from 49² – 43², 2(49)(43), 49² + 43²

That gives us 2 more triples to add to the previous 8 for a total of 10 Pythagorean triples, and my conjecture still holds true.

Now one more thing about the number 850, here’s how to write it in a couple other bases:

  • 505 BASE 13, because 5(13²) + 5(1) = 5(170) = 850.
  • PP BASE 33 (P is 25 base 10) because 25(33) + 25(1) = 25(34) = 850

 

 

824 and Level 3

824 is the sum of all the prime numbers from 61 all the way to 103, which just happens to be one of its prime factors!

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

824 is a leg in a few Pythagorean triples:

  • 618-824-1030 because that is 206 times (3-4-5)
  • 824-1545-1751 because that is 103 times (8-15-17)
  • 824-10593-10625 because 2(103)(4) = 824
  • 824-21210-21226 because 105² – 101² = 824
  • 824-42432-42440 because 2(206)(2) = 824
  • 824-84870-84874 because 207² – 205² = 824
  • Primitive 824-169743-169745 because 2(412)(1) = 824

Five of those triples were derived directly from 824’s factor pairs.

Two of the triples were derived indirectly:

  • What is (105+101)/2, (105-101)/2?
  • Also, what is (207+205)/2, (207+205)/2?

The answer to both questions is a factor pair of 824.

You can read more about finding Pythagorean triples for numbers that are divisible by 4 here.

  • 824 is a composite number.
  • Prime factorization: 824 = 2 × 2 × 2 × 103, which can be written 824 = 2³ × 103
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 824 has exactly 8 factors.
  • Factors of 824: 1, 2, 4, 8, 103, 206, 412, 824
  • Factor pairs: 824 = 1 × 824, 2 × 412, 4 × 206, or 8 × 103
  • Taking the factor pair with the largest square number factor, we get √824 = (√4)(√206) = 2√206 ≈ 28.7054

 

 

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

 

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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