A Multiplication Based Logic Puzzle

Archive for August, 2016

780 is the 39th Triangular Number

  • 780 is a composite number.
  • Prime factorization: 780 = 2 x 2 x 3 x 5 x 13, which can be written 780 = (2^2) x 3 x 5 x 13
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 3 x 2 x 2 = 24. Therefore 780 has exactly 24 factors.
  • Factors of 780: 1, 2, 3, 4, 5, 6, 10, 12, 13, 15, 20, 26, 30, 39, 52, 60, 65, 78, 130, 156, 195, 260, 390, 780
  • Factor pairs: 780 = 1 x 780, 2 x 390, 3 x 260, 4 x 195, 5 x 156, 6 x 130, 10 x 78, 12 x 65, 13 x 60, 15 x 52, 20 x 39, or 26 x 30
  • Taking the factor pair with the largest square number factor, we get √780 = (√4)(√195) = 2√195 ≈ 27.92848.

780-factor-pairs

There are MANY ways to make factor trees for 780. Here are just three of them:

780 Factor Trees

Stetson.edu informs us that (7 + 5)(8 + 5)(0 + 5) = 780.

780 is the sum of consecutive prime numbers two different ways:

  • 59 + 61 + 67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 = 780 (10 consecutive primes).
  • 191 + 193 + 197 + 199 = 780 (4 of my favorite consecutive primes).

26 and 30 are both exactly 2 numbers away from their average 28, so 780 can be written as the difference of two squares:

780 = 26 x 30 = (28 – 2)(28 + 2) = 28² – 2² = 784 – 4.

(It can be written as the difference of two squares three other ways, but I won’t list them here.)

Because 5 and 13 are both factors of 780, it is the hypotenuse of FOUR Pythagorean triangles making each of these equations true:

  • 192² + 756² = 780²
  • 300² + 720² = 780²
  • 396² + 672² = 780²
  • 468² + 624² = 780²

780 can be written as the sum of three perfect squares two different ways:

  • 26² + 10² + 2² = 780
  • 22² + 14² + 10² = 780

780 BASE 10 is palindrome QQ BASE 29. (Q is 26 in BASE 10.) Note that 26(29) + 26(1) = 780.

1 + 2 + 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 + 39 = 780, making 780 the 39th triangular number. Since 20 is in the exact middle of that list of numbers, 780 is also the 20th hexagonal number.

Triangular numbers are interesting, but are they good for anything? Here’s one good thing:

Count the Terms of Sums Squared

There are 26 letters in the English alphabet. 39 variables would use the alphabet exactly one and one half times, but that’s okay because we can use upper case letters the second time around:

(a + b + c + . . . + x + y+ z + A + B + C + . . . + K + L + M)² has exactly 780 terms because it has 39 single variables, and 780 is the 39th triangular number.

780 is the 39th triangular number because 39⋅40/2 = 780.

A couple of months ago I saw a fascinating image on twitter. It is a square made with 1001 dots representing the numbers from -500 to +500 with zero in the exact middle. The triangular numbers are represented by the “+” pattern seen here. If we added about 600 more dots to the square, -780 and +780 would also lie on that +.

//platform.twitter.com/widgets.js

Half of all triangular numbers are also hexagonal numbers. If there were enough dots, then -780 and +780 would also be among the dots forming the “-” pattern representing the hexagonal numbers. 780 is the 20th hexagonal number because 20(2⋅20-1) = 780.

 

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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

 

 

 

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