A Multiplication Based Logic Puzzle

  • 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.

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 gif 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

If you start the gif, you will see a different pattern formed by the square numbers, the pentagonal number, and so forth up to the nonagonal numbers.

Half of all triangular numbers are also hexagonal numbers. If the gif had 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.

 

Before reading about the number 779 further down in this post, try finding the factors from 1-10 that make puzzle #779 function as a multiplication table. It’s a level 5 puzzle so it can be a bit tricky.

779 Puzzle

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  • 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.

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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

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And finally a little 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.

In a future post I’ll show how Pythagorean triples can be generated when some of the factors are even.

 

 

778 and Level 4

Stetson.edu informs us that 778 is the number of ways  a 1 x 5 rectangle can be surrounded by other 1 x 5 rectangles. I’m not going to publish or even find all 778 possible ways, but here is one of them:

778 Surround

27² + 7² = 778.

We can use those numbers to compute a Pythagorean triple: 2(27)(7), 27² – 7², 27² + 7², which turns out to be 378-680-778, and thus 378² + 680² = 778².

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

  • 25² + 12² + 3² = 778
  • 24² + 11² + 9² = 778
  • 21² + 16² + 9² = 778

778 is palindrome 1G1 BASE 21 (G is 16 base 10)

That palindrome means that 1(21²) + 16(21) + 1(1) = 778.

778 Puzzle

  • 778 is a composite number.
  • Prime factorization: 778 = 2 x 389
  • 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 778 has exactly 4 factors.
  • Factors of 778: 1, 2, 389, 778
  • Factor pairs: 778 = 1 x 778 or 2 x 389
  • 778 has no square factors that allow its square root to be simplified. √778 ≈ 27.892651.

778 Logic

Some people think that 7 is a lucky number. If that is true, then 777 should be even luckier.

Some numbers are lucky enough to be included in Multiplication Rhymes.ppt – mathval, a fun power point that helps students learn 12 multiplication facts, including these three that use Lucky Numbers:

  • 3 & 7 are always lucky numbers; 3 x 3 = 9 lives of a cat.
  • 3 & 7 are always lucky numbers; 3 x 7 = 21 lucky age.
  • 3 & 7 are always lucky numbers; 7 x 7 = 49er Gold Miner.

In Number Theory Lucky Numbers are actually defined and can be generated using a sieve somewhat similar to the prime number generating Sieve of Eratosthenes. There is an infinite number of Lucky Numbers, and yes, 3, 7, 9, 21, 49, and 777 all make the list.

Oeis.org’s wiki, Lucky numbers, includes several lucky number lists including the first 33 composite Lucky Numbers thus defined because ALL of their factors are also Lucky Numbers. 777 was the 19th number on that particular list because ALL of its factors, 1, 3, 7, 21, 37, 111, 259, 777 are Lucky Numbers, too!

If that isn’t lucky enough, 777 is a repdigit in three different bases.

  • 3333 BASE 6; note that 3(6^3) + 3(6^2) + 3(6^1) + 3(6^0) = 777
  • 777 BASE 10; note that 7(100) + 7(10) + 7(1) = 777
  • LL BASE 36 (L is 21 base 10); note that 21(36) + 21(1) = 777

Did you notice that lucky numbers 3, 7, and 21 showed  up again? I liked that coincidence so much that I made this graphic:

777 Repdigit

777 is also the sum of three squares four different ways:

  • 26² + 10² + 1² = 777
  • 22² + 17² + 2² = 777
  • 20² + 19² + 4² = 777
  • 20² + 16² + 11² = 777

Now here is the factoring information for the number 777:

  • 777 is a composite number.
  • Prime factorization: 777 = 3 x 7 x 37
  • 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 777 has exactly 8 factors.
  • Factors of 777: 1, 3, 7, 21, 37, 111, 259, 777
  • Factor pairs: 777 = 1 x 777, 3 x 259, 7 x 111, or 21 x 37
  • 777 has no square factors that allow its square root to be simplified. 77727.8747197.

776 and Level 3

776 is the sum of two squares: 26² + 10² =776.

776 is the hypotenuse of Pythagorean triple 520-576-776 so 520² + 576² = 776².

776 is also the sum of three squares five different ways:

  • 26² + 8² + 6² = 776
  • 24² + 14² + 2² = 776
  • 24² + 10² + 10² = 776
  • 22² + 16² + 6² = 776
  • 18² + 16² + 14² = 776

776 is a palindrome in three other bases:

  • 646 BASE 11; note that 6(121) + 4(11) + 6(1) = 776
  • 272 BASE 18; note that 2(18²) + 7(18) + 2(1) = 776
  • 161 BASE 25; note that 1(25²) + 6(25) + 1(1) = 776

776 Puzzle

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  • 776 is a composite number.
  • Prime factorization: 776 = 2 x 2 x 2 x 97, which can be written 776 = (2^3) x 97
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 776 has exactly 8 factors.
  • Factors of 776: 1, 2, 4, 8, 97, 194, 388, 776
  • Factor pairs: 776 = 1 x 776, 2 x 388, 4 x 194, or 8 x 97
  • Taking the factor pair with the largest square number factor, we get √776 = (√4)(√194) = 2√194 ≈ 27.85677655

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776 Factors

775 and Level 2

775 is part of several Pythagorean triples including two that are primitive:

  • 168-775-793 (Primitive)
  • 775-1860-2015
  • 775-9672-9703
  • 775-12000-12025
  • 775-300312-300313 (Primitive)

775 is palindrome PP in BASE 30 (P = 25 base 10). Note that 25(30) +25(1) = 775.

775 is also the sum of three triangular numbers 9 different ways:

  1. 3+276+ 496
  2. 6+28+741
  3. 6+66+703
  4. 10+300+465
  5. 15-325-435
  6. 36+36+703
  7. 78+136+561
  8. 171+253+351
  9. 120+190+465

775 Puzzle

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  • 775 is a composite number.
  • Prime factorization: 775 = 5 x 5 x 31, which can be written 775 = (5^2) x 31
  • 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 775 has exactly 6 factors.
  • Factors of 775: 1, 5, 25, 31, 155, 775
  • Factor pairs: 775 = 1 x 775, 5 x 155, or 25 x 31
  • Taking the factor pair with the largest square number factor, we get √775 = (√25)(√31) = 5√31 ≈ 27.83882181.

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775 Factors

774 and Level 1

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

  • 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 = 774; that’s 18 consecutive primes.
  • 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 774; that’s 14 consecutive primes.

774 is the sum of three triangular numbers eight different ways:

  • 666 + 105 + 3 = 774
  • 630 + 78 + 66 = 774
  • 561 + 210 + 3 = 774
  • 528 + 231 + 15 = 774
  • 528 + 210 + 36 = 774
  • 465 + 231 + 78 = 774
  • 378 + 351 + 45 = 774
  • 378 + 276 + 120 = 774

774 is also the sum of three squares nine different ways:

  • 27² + 6² + 3² = 774
  • 26² + 7² + 7² = 774
  • 25² + 10² + 7² = 774
  • 23² + 14² + 7² = 774
  • 22² + 17² + 1² = 774
  • 22² + 13² + 11² = 774
  • 21² + 18² + 3² = 774
  • 18² + 15² + 15² = 774
  • 17² + 17² + 14² = 774

774 Puzzle

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  • 774 is a composite number.
  • Prime factorization: 774 = 2 x 3 x 3 x 43, which can be written 774 = 2 x (3^2) x 43
  • 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 774 has exactly 12 factors.
  • Factors of 774: 1, 2, 3, 6, 9, 18, 43, 86, 129, 258, 387, 774
  • Factor pairs: 774 = 1 x 774, 2 x 387, 3 x 258, 6 x 129, 9 x 86, or 18 x 43
  • Taking the factor pair with the largest square number factor, we get √774 = (√9)(√86) = 3√86 ≈ 27.820855.

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774 Factors

 

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