A Multiplication Based Logic Puzzle

Archive for the ‘Level 5 Puzzle’ Category

804 Is There a Pot of Gold at the End of This Rainbow?

Factor Rainbows can be a wonderful way to display the factors of a number. Not only are all the factors listed in order from smallest to greatest, but the factor pairs are joined together with the same color band.

The number 804 has 12 factors so it makes a lovely rainbow with 6 different color bands.

Is there a pot of gold at the end of this factor rainbow? I’ll let you decide the answer to that question.

Print the puzzles or type the solution on this excel file: 10-factors 801-806

Finding golden nuggets of information about a number might be less difficult than finding pots of gold.

I always begin the painstaking mining process by looking at the factors of the number:

  • 804 is a composite number.
  • Prime factorization: 804 = 2 x 2 x 3 x 67, which can be written 804 = (2^2) x 3 x 67
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 804 has exactly 12 factors.
  • Factors of 804: 1, 2, 3, 4, 6, 12, 67, 134, 201, 268, 402, 804
  • Factor pairs: 804 = 1 x 804, 2 x 402, 3 x 268, 4 x 201, 6 x 134, or 12 x 67
  • Taking the factor pair with the largest square number factor, we get √804 = (√4)(√201) = 2√201 ≈ 28.3548937575

About half of everything there was already in the factor rainbow.

Finding nuggets of information about the number 804 has been a little difficult and disappointing:

  • None of 804’s prime factors can be written as 4N+1, so 804 is NOT the hypotenuse of any Pythagorean triples.
  • 804 is NOT a palindrome in base 36 or any base less than that.
  • 804 is NOT the sum of any consecutive prime numbers.

Even though I did not find any golden nuggets in those places, I kept looking and finally found a couple of gems about the number 804:

804 can be written as the sum of three squares four different ways, and all of those ways have some definition of double in them:

  • 28² + 4² + 2² = 804
  • 26² + 8² + 8² = 804
  • 22² + 16² + 8² = 804
  • 20² + 20² + 2² = 804

Stetson.edu also gives us a nugget about the number 804 that may be a bit too heavy for most people to handle: “804 is a value of n for which 2φ(n) = φ(n+1).” That basically means that there are exactly half as many numbers less than 804 that are NOT divisible by its prime factors (2, 3, or 67) as there are numbers less than 805 that are NOT divisible by its prime factors (5, 7, or 23).

I started looking for golden specs about 804 in places that I don’t usually look.

267 + 268 + 269 = 804 so 804 is the sum of 3 consecutive numbers.

As stated before 804 is never the hypotenuse of a Pythagorean triple. However to find all the times it is a leg in a triple will require a lot of labor especially since 804 has so many factors, including 4, and two of its factor pairs have factors where both factors are even.

  • 134 × 6 is an even factor pair, so (134 + 6)/2 = 70, and (134-6)/2 = 64. Thus 804 = 134·6 = (70 + 64)(70 – 64) = 70² – 64² .
  • 402 × 2 is another even factor pair, so (402 + 2)/2 = 202, and (402 – 2)/2 = 200. Thus 804 = 402·2 = (200 + 2)(200 – 2) = 202² – 200²
  • Likewise odd or even sets of factor pairs of any of 804’s factors can also be used to find Pythagorean triples.

So to find all Pythagorean triples that contain the number 804, we will have to find all the times 804 satisfies one of these FOUR conditions:

  1. 804 = 2k(a)(b) so that 804 is in the triple 2k(a)(b), k(a² – b²), k(a + b²) OR the triple k(a² – b²), 2k(a)(b), k(a + b²).
  2. 804 = 2(a)(b) so that 804 is in the triple 2(a)(b), a² – b², a + b² OR the triple a² – b², 2(a)(b), a + b².
  3. 804 = a² – b² so that 804 is in the triple a² – b², 2(a)(b), a + b² OR the triple 2(a)(b), a² – b², a + b².
  4. 804 = k(a² – b²) so that 804 is in the triple k(a² – b²), 2k(a)(b), k(a + b²) OR the triple 2k(a)(b), k(a² – b²), k(a + b²).

Let the mining process begin! I’ll list the triples with the shortest legs first and color code each triple according to the condition I used.

  • 335-804-871 which used 804 = 2·67(3)(2) to make a triple that is 5-12-13 times 67
  • 603-804-1005 which used 804 = 2·201(2)(1) to make a triple that is 3-4-5 times 201
  • 804-1072-1340 which used 804 = 268(2² – 1²) to make a triple that  is 3-4-5 times 268
  • 804-2345-2479 which used 804 = 2·67(6)(1) to make a triple that is 12-35-37 times 67
  • 804-4453-4525, which used 804 = 2(6)(67)
  • 804-8960-8996, which used 804 = 70² – 64² or 804 = 4(35² – 32²) to make a triple that is 201-2240-2249 times
  • 804-17947-17965, which used 804 = 2(134)(3)
  • 804-26928-26940 which used 804 = 12(34² – 33²) to make a triple that is 67-2244-2245 times 12
  • 804-40397-40405, which used 804 = 2(201)(2)
  • 804-53865-53871 which used 804 = 2·3(134)(1) to make a triple that is 268-17955-17957 times 3
  • 804-80800-80804 which used 804 = 202² – 200²  or 804 = 4(101² – 100²) to make a triple that is 201-20200-20201 times 4
  • 804-161603-161605, a primitive Pythagorean triple, that used 804 = 2(402)(1)

If you look for a pot of gold at the end of a rainbow, you’re bound to be disappointed. Science/How Stuff Works just had to crush dreams and dispel 10 Myths About Rainbows. Unfortunately a pot of gold being at the rainbow’s end is included on that list. Still I suppose we could still put every golden spec or nugget about 804 into a little pot and call it a pot of gold.

Or if you are as clever and quick as a leprechaun, perhaps you will consider finding Pythagorean triples to be like finding pots of gold.

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798 Cupid’s Arrow and Target

Here are two puzzles that go together and yet look out of sync. Sometimes cupid’s arrow reaches its target, and sometimes it doesn’t.

798-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-795-799

Here’s a little about the number 798:

798 is made from three consecutive numbers (7, 8, and 9), so it is divisible by three. The middle number, 8, is not divisible by three, so 798 is NOT divisible by nine.

798 is a palindrome in two bases:

  • 666 BASE 11 because 6(121) + 6(11) + 6(1) = 798
  • 383 BASE 15 because 3(225) + 8(15) + 3(1) = 798

798 is also the sum of two consecutive prime numbers: 397 + 401 = 798.

798 can be written as the sum of three squares four different ways:

  • 26² + 11² + 1² = 798
  • 25² + 13² + 2² =798
  • 23² + 13² + 10² = 798
  • 22² + 17² + 5² = 798

Here is 798’s factoring information:

  • 798 is a composite number.
  • Prime factorization: 798 = 2 x 3 x 7 x 19
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 798 has exactly 16 factors.
  • Factors of 798: 1, 2, 3, 6, 7, 14, 19, 21, 38, 42, 57, 114, 133, 266, 399, 798
  • Factor pairs: 798 = 1 x 798, 2 x 399, 3 x 266, 6 x 133, 7 x 114, 14 x 57, 19 x 42, or 21 x 38
  • 798 has no square factors that allow its square root to be simplified. √798 ≈ 28.24889.

798-factor-pairs

793 and Level 5

793 is the sum of two squares TWO different ways!

  • 28² + 3² = 793
  • 27² + 8² = 793

Notice that 793 is 4(198) + 1, and neither 28 and 3 or 27 and 8 have any common prime factors. Could 793 possibly be a prime number?

The answer is no for two reasons:

  1. √793 is about 28.2, so we only need to check to see if 793 is divisible by 5, 13, or 17 (all the 4N+1 prime numbers that are less than 28.) However, 793 ÷ 13 = 61, so  793 isn’t prime.
  2. Also, any number that can be written as the sum of two squares in more than one way is never a prime number.

Both 13 and 61 give a remainder of one when they are divided by four, and 793 is the hypotenuse of FOUR Pythagorean triples:

  • 143-780-793, which is 13 times 11-60-61
  • 168-775-793, a primitive calculated from 2(28)(3), 28² – 3², 28² + 3²
  • 305-732-793, which is 61 times 5-12-13
  • 432-665-793, a primitive calculated from 2(27)(8), 27² – 8², 27² + 8²

793 is also palindrome 191 in BASE 24; note that 1(24²) + 9(24) + 1(1) = 793

Finally Stetson.edu informs us that 793 is 2(397) – 1.

793-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

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  • 793 is a composite number.
  • Prime factorization: 793 = 13 x 61
  • 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 793 has exactly 4 factors.
  • Factors of 793: 1, 13, 61, 793
  • Factor pairs: 793 = 1 x 793 or 13 x 61
  • 793 has no square factors that allow its square root to be simplified. √793 ≈ 28.16025568.

793-factor-pairs

786 and Level 5

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

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

 

 

 

772 and Level 5

  • 772 is a composite number.
  • Prime factorization: 772 = 2 x 2 x 193, which can be written 772 = (2^2) x 193
  • 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 772 has exactly 6 factors.
  • Factors of 772: 1, 2, 4, 193, 386, 772
  • Factor pairs: 772 = 1 x 772, 2 x 386, or 4 x 193
  • Taking the factor pair with the largest square number factor, we get √772 = (√4)(√193) = 2√193 ≈ 27.78488798.

772-factor-pairs

Here is a factoring puzzle to try:

772 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-02-25

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Here’s more about the number 772:

24² + 14² = 772 so 772 is the hypotenuse of a Pythagorean triple, and 380² + 672² = 772². 380-672-772 is calculated from 24² – 14², 2(24)(14) , 24² + 14².

22² + 12² + 12² = 772, making 772 the sum of three square numbers.

772 is also the sum of two consecutive prime numbers: 383 + 379 = 772.

Stetson.edu informs us that 772 is the smallest number that is the sum of three triangular numbers 21 different ways. I decided to find all those ways for myself and share them here. (If zero wasn’t named the zeroth triangular number, there would “only” be 20 ways.)

772 Smallest Number That

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772 Logic


 

Reducible Square Roots up to √765

  • 765 is a composite number.
  • Prime factorization: 765 = 3 x 3 x 5 x 17, which can be written 765 = (3^2) x 5 x 17
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 765 has exactly 12 factors.
  • Factors of 765: 1, 3, 5, 9, 15, 17, 45, 51, 85, 153, 255, 765
  • Factor pairs: 765 = 1 x 765, 3 x 255, 5 x 153, 9 x 85, 15 x 51, or 17 x 45
  • Taking the factor pair with the largest square number factor, we get √765 = (√9)(√85) = 3√85 ≈ 27.658633.

765-factor-pairs

765 is the 300th number whose square root can be reduced! Here are three tables with 100 reducible square roots each showing all the reducible square roots up to √765. When three or more consecutive numbers have reducible square roots, I highlighted them.

1st 100 reducible square roots

2nd 100 reducible square roots

Reducible Square Roots 516-765

That’s 300 reducible square roots found for the first 765 counting numbers. 300 ÷ 765 ≈ 0.392, so 39.2% of the numbers so far have reducible square roots.

Today’s puzzle is a whole lot less complicated than all that, so give it a try!

765 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2016-02-04

Logical steps to find the solution are in a table at the bottom of the post.

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Here are some other fun facts about the number 765:

765 is made from three consecutive numbers so it is divisible by 3. The middle of those numbers is 6 so 765 is also divisible by 9.

765 can be written as the sum of two squares two different ways:

  • 27² + 6² = 765
  • 21² + 18² = 765

Its other two prime factors, 5 and 17, have a remainder of 1 when divided by 4 so 765² can be written as the sum of two squares FOUR different ways, two of which contain other numbers that use the same digits as 765. Also notice that 9 is a factor of each number in the corresponding Pythagorean triples.

  • 117² + 756² = 765²
  • 324² + 693² = 765²
  • 360² + 675² = 765²
  • 459² + 612² = 765²

765 can also be written as the sum of three squares four different ways:

  • 26² + 8² + 5² = 765
  • 22² + 16² + 5² = 765
  • 20² + 19² + 2² = 765
  • 20² + 14² + 13² = 765

765 is a palindrome in two different bases:

  • 1011111101 BASE 2; note that 1(512) + 0(256) + 1(128) + 1(64) + 1(32) + 1(16) + 1(8) + 1(4) + 0(2) + 1(1) = 765.
  • 636 BASE 11; note that 6(121) + 3(11) + 6(1) = 765.

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765 Logic

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