A Multiplication Based Logic Puzzle

Archive for June, 2016

778 and Level 4

  • 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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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.

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778 Logic

How Lucky Can 777 Be?

  • 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.

777-factor-pairs

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

 

776 and Level 3

  • 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

775 and Level 2

  • 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.

775-factor-pairs

Here’s today’s factoring puzzle:

775 Puzzle

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

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Here is more about the number 775:

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

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775 Factors

 


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