Puzzles

787 Always a Unique Solution

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  • 787 is a prime number.
  • Prime factorization: 787 is prime.
  • The exponent of prime number 787 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 787 has exactly 2 factors.
  • Factors of 787: 1, 787
  • Factor pairs: 787 = 1 x 787
  • 787 has no square factors that allow its square root to be simplified. √787 ≈ 28.05352.

787-factor-pairs

Now for today’s puzzle….The fact that these Find the Factor puzzles always have a unique solution is an important clue in solving this rather difficult puzzle. Good luck!

787-puzzle

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

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Here’s more about the number 787:

787 is a palindrome in bases 4, 10, 11 and 16:

  • 30103 BASE 4; note that 3(256) + 0(64) + 1(16) + 0(4) + 3(1) = 787
  • 787 BASE 10; note that 7(100) + 8(10) + 7(1) = 787
  • 656 BASE 11; note that 6(121) + 5(11) + 6(1) = 787
  • 313 BASE 16; note that 3(256) + 1(16) + 3(1) = 787

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What did I mean when I wrote that the puzzles always having a unique solution is an important clue? There is only one clue in the puzzle that is divisible by 11. One of the rows and one of the columns do not have a clue, so the other 11 will go with one of them. The cell where the empty row and empty column intersect cannot be 132 because if that worked, it would produce two possible solutions to the puzzle. This table explains a logical order to find the solution.

787-logic

786 and Level 5

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786 is even so it is divisible by 2. Also since 786 is made from 3 consecutive numbers, we can tell automatically that it is divisible by 3. Those two facts together mean 786 is also divisible by 6.

  • 786 is a composite number.
  • Prime factorization: 786 = 2 x 3 x 131
  • 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 786 has exactly 8 factors.
  • Factors of 786: 1, 2, 3, 6, 131, 262, 393, 786
  • Factor pairs: 786 = 1 x 786, 2 x 393, 3 x 262, or 6 x 131
  • 786 has no square factors that allow its square root to be simplified. √786 ≈ 28.03569.

786-factor-pairs

Today’s Find the Factors puzzle:

786-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 786:

786 is 123 in BASE 27 because 1(27²) + 2(27) + 3(1) = 786.

786 is the sum of two consecutive primes: 389 + 397 = 786

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786-logic

785 and Level 4

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  • 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

 

784 and Level 3

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If you know the multiplication facts up to 12 x 12, then it is obvious that 784 is divisible by 7. However, 784 is divisible by a whole lot more numbers than that. 784 can actually be evenly divided by 15 different numbers. Hmm, 15, that’s an odd number. A number’s factors always come in pairs. One of its factor pairs must contain the same factor twice, meaning 784 is a perfect square!

  • 784 is a composite number.
  • Prime factorization: 784 = 2 x 2 x 2 x 2 x 7 x 7, which can be written 784 = (2^4) x (7^2)
  • The exponents in the prime factorization are 4 and 2. Adding one to each and multiplying we get (4 + 1)(2 + 1) = 5 x 3 = 15. Therefore 784 has exactly 15 factors.
  • Factors of 784: 1, 2, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392, 784
  • Factor pairs: 784 = 1 x 784, 2 x 392, 4 x 196, 7 x 112, 8 x 98, 14 x 56, 16 x 49, or 28 x 28
  • 784 is a perfect square. √784 = 28

784-factor-pairs

But that’s not the only thing remarkable about this perfect square: √784 is 28, the 7th triangular number, so like all other squared triangular numbers 784 has this additional property:

784-sum-of-consecutive-cubes

Just as 784 is a perfect square, five of the twelve clues in today’s puzzle are also perfect squares. But don’t let that fact trick you into writing the same factor in both the first column and the top row every time!

784-puzzle

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

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Here’s more about the number 784:

784 can be written as the sum of three squares in exactly one way:

  • 24² + 12² + 8² = 784

784 is also a palindrome in Bases 13, 17, and 27:

  • 484 BASE 13; note that 4(169) + 8(13) + 4(1) = 784
  • 2C2 BASE 17 (C is 12 base 10); note that 2(289) + 12(17) + 2(1) = 784
  • 121 BASE 27; note that 1(27²) + 2(27) + 1(1) = 784

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784-factors

 

 

783 and Level 2

/ ivasallay / Leave a comment
  • 783 is a composite number.
  • Prime factorization: 783 = 3 x 3 x 3 x 29, which can be written 783 = (3^3) x 29
  • 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 783 has exactly 8 factors.
  • Factors of 783: 1, 3, 9, 27, 29, 87, 261, 783
  • Factor pairs: 783 = 1 x 783, 3 x 261, 9 x 87, or 27 x 29
  • Taking the factor pair with the largest square number factor, we get √783 = (√9)(√87) = 3√87 ≈ 27.982137.

783-factor-pairs

Here’s today’s puzzle. It’s a level 2 so it isn’t very difficult:

783-puzzle

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

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27 x 29 = 783. Since (n – 1)(n + 1) always equals n² – 1, we know that 783 is one number away from the next perfect square.

29 is a factor of 783, making 783 the hypotenuse of a Pythagorean triple:

  • 540-567-783, which is 27 times 20-21-29.
  • Thus 540² + 567² = 783² just as 20² + 21² = 29².

783 is also a palindrome in bases 15, 23, and 28:

  • 373 BASE 15; note that 3(225) + 7(15) + 3(1) = 783
  • 1B1 BASE 23 (B is 11 base 10); note that 1(23²) + 11(23) + 1(1) = 783
  • RR BASE 28 (R is 27 base 10); note that 27(28) + 27 = 783

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783-factors

782 and Level 1

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  • 782 is a composite number.
  • Prime factorization: 782 = 2 x 17 x 23
  • 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 782 has exactly 8 factors.
  • Factors of 782: 1, 2, 17, 23, 34, 46, 391, 782
  • Factor pairs: 782 = 1 x 782, 2 x 391, 17 x 46, or 23 x 34
  • 782 has no square factors that allow its square root to be simplified. √782 ≈ 27.96426.

782-factor-pairs

Here’s today’s puzzle:

782-puzzle

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

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Here’s some more information about the number 782:

OEIS.org informs us that the sum of the factors of 782 is not only a perfect square number but also a perfect 4th power.

Specifically, 1 + 2 + 17 + 23 + 34 + 46 + 391 + 782 = 1296, which happens to be 6^4.

23 is one of the factors of 782, and here are three interesting number facts about the number 782 that involve the number 23:

  • 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 = 782. (That’s 23 consecutive numbers beginning with the number 23.)
  • 782 is the 23rd pentagonal number because 23(3⋅23 – 1)/2 = 782.
  • 782 is palindrome NN in BASE 33. (N is 23 base 10.) Note that 23(33) + 23(1) = 782

Because 17 is one of its factors, 782 is the hypotenuse of Pythagorean triple 368-690-782.

Thus, 368² + 690² = 782².

Also since 17 x 46 = 782, the greatest common factor of 368, 690 and 782 is 46.

782 is also the sum of three squares six different ways:

  • 27² + 7² + 2² = 782
  • 26² + 9² + 5² = 782
  • 25² + 11² + 6² = 782
  • 22² + 17² + 3² = 782
  • 19² + 15² + 14² = 782
  • 18² + 17² + 13² = 782

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782-factors

781 and Level 6

/ ivasallay / Leave a comment
  • 781 is a composite number.
  • Prime factorization: 781 = 11 x 71
  • 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 781 has exactly 4 factors.
  • Factors of 781: 1, 11, 71, 781
  • Factor pairs: 781 = 1 x 781 or 11 x 71
  • 781 has no square factors that allow its square root to be simplified. √781 ≈ 27.94637722.

781-factor-pairs

Can you solve today’s puzzle?

781-puzzle

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

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Here’s more about the number 781:

781 is the sum of the 19 prime numbers from 7 to 79.

Thus 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 = 781

781 is a repdigit in base 5 and a palindrome in three other bases:

  • 11111 BASE 5. Note that 625 + 125 + 25 + 5 + 1 = 781.
  • 232 BASE 19. Note that 2(19²) + 3(19) + 2(1) = 781
  • 1J1 BASE 20, J = 19 base 10. Note that 1(20²) + 19(20) + 1(1) = 781
  • 141 BASE 26. Note that 1(26²) + 4(26) + 1(1) = 781

781 is also the sum of three squares five ways

  • 27² + 6² + 4² = 781
  • 24² + 14² + 3² = 781
  • 24² + 13² + 6² = 781
  • 21² + 18² + 4² = 781
  • 21² + 14² + 12² = 781

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781-logic

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

/ ivasallay / 2 Comments
  • 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

 

 

 

778 and Level 4

/ ivasallay / 1 Comment
  • 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-factor-pairs

Here’s today’s puzzle. Hints for solving it are in a table at the end of the post.

 

778 Puzzle

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

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OEIS.org 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.

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778 Logic

776 and Level 3

/ ivasallay / Leave a comment
  • 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

776-factor-pairs

Try solving today’s puzzle:

776 Puzzle

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

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Here’s more about the number 776:

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

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776 Factors