Level 6 Puzzle

701 Some Virgács left by Mikulás

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  • 701 is a prime number.
  • Prime factorization: 701 is prime.
  • The exponent of prime number 701 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 701 has exactly 2 factors.
  • Factors of 701: 1, 701
  • Factor pairs: 701 = 1 x 701
  • 701 has no square factors that allow its square root to be simplified. √701 ≈ 26.4764046.

How do we know that 701 is a prime number? If 701 were not a prime number, then it would be divisible by at least one prime number less than or equal to √701 ≈ 26.4. Since 701 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 701 is a prime number.

Tonight Mikulás will visit the homes of children who sleep in Hungary. If they have been good, he will fill their boots with sweet treasures. If they have been naughty, they will receive virgács, small twigs that have been spray painted gold and bound together with red decorative ribbon. Actually most children make both the naughty list and the nice list so their boots are filled with a mixture of sweet and the not so sweet including virgács, a subtle reminder to be good.

I especially like this illustration from Wikipedia that features Mikulás (Saint Nickolas) and Krampusz:

I like that it is 150 years old. It is from 1865, several decades before any of my husband’s grandparents left Hungary to live in the United States. Under the chair is a little boy hiding from Krampusz. I like to imagine he’s related to my husband some way. The little girl in the illustration must have been much better behaved that year because she is not afraid enough to need to hide.

Since everyone has been at least a little bit naughty this year, here is virgács for you to put in your shoes tonight, too.

701 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-30

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Here is a little more about the number 701:

26² + 5² = 701 so it is the hypotenuse of the primitive Pythagorean triple 260-651-701 which can be calculated using 2(26)(5), 26² – 5², 26² + 5².

701 is the sum of three consecutive prime numbers: 229 + 233+ 239.

701 is a palindrome in several bases:

  • 10301 BASE 5; note that 1(625) + 0(125) + 3(25) + 0(5) + 1(1) = 701.
  • 858 BASE 9; note that 8(81) + 5(9) + 8(1) = 701.
  • 1F1 BASE 20; note that F is equivalent to 15 in base 10, and 1(400) + 15(20) + 1(1) = 701.
  • 131 BASE 25; note that 1(625) + 3(25) + 1(1) = 701

OEIS.org informs us that 1^0 + 2^1 + 3^2 + 4^3 + 5^4 = 701.

Here’s another way we know that 701 is a prime number: Since 701 ÷ 4 has a remainder of 1, and 701 can be written as the sum of two squares that have no common prime factors (26² + 5² = 701), then 701 will be prime unless it is divisible by a primitive Pythagorean hypotenuse less than or equal to √701 ≈ 26.4. Since 701 is not divisible by 5, 13, or 17, we know that 701 is a prime number.

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701 Logic

694 and Level 6

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  • 694 is a composite number.
  • Prime factorization: 694 = 2 x 347
  • 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 694 has exactly 4 factors.
  • Factors of 694: 1, 2, 347, 694
  • Factor pairs: 694 = 1 x 694 or 2 x 347
  • 694 has no square factors that allow its square root to be simplified. √694 ≈ 26.34388.

Level 6 puzzles aren’t easy, but do give it a try! There are hints at the bottom of the page.

694 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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Here are some ways that 694 is a special number:

694 is the 22nd centered triangular number because it is the sum of these consecutive triangular numbers 210, 231, 253.

Notice that 21(22)/2 is 231, the number in the middle of those consecutive triangular numbers. Also notice that 3(231) + 1 = 694.

694 BASE 10 becomes palindrome 1C1 in BASE 21 because C Base 21 is equivalent to 12 Base 10, and 1(21²) + 12(21) + 1(1) = 694.

 

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694 Logic

687 and Level 6

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687 is made from three consecutive numbers so it is divisible by 3.

687 is the hypotenuse of Pythagorean triple 180-663-687. What is the greatest common factor of those three numbers?

687 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-16

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  • 687 is a composite number.
  • Prime factorization: 687 = 3 x 229
  • 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 687 has exactly 4 factors.
  • Factors of 687: 1, 3, 229, 687
  • Factor pairs: 687 = 1 x 687 or 3 x 229
  • 687 has no square factors that allow its square root to be simplified. √687 ≈ 26.21068.

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687 Logic

Ricardo tweeted the solution for this one:

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679 and Level 6

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679 is the sum of the three primes from 223 to 229 and the sum of the nine primes from 59 to 97.

Since 97 is one of its factors, 679 is the hypotenuse of the Pythagorean triple 455-504-679. What is the greatest common factor of those three numbers?

679 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-09

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  • 679 is a composite number.
  • Prime factorization: 679 = 7 x 97
  • 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 679 has exactly 4 factors.
  • Factors of 679: 1, 7, 97, 679
  • Factor pairs: 679 = 1 x 679 or 7 x 97
  • 679 has no square factors that allow its square root to be simplified. √679 ≈ 26.057628.

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679 Logic

673 and Level 6

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673 = 23² + 12².

673 is the hypotenuse of the primitive Pythagorean triple 385-552-673 which was calculated using 23² – 12², 2(23)(12), 23² + 12².

673 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-02

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  • 673 is a prime number.
  • Prime factorization: 673 is prime.
  • The exponent of prime number 673 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 673 has exactly 2 factors.
  • Factors of 673: 1, 673
  • Factor pairs: 673 = 1 x 673
  • 673 has no square factors that allow its square root to be simplified. √673 ≈ 25.94224.

How do we know that 673 is a prime number? If 673 were not a prime number, then it would be divisible by at least one prime number less than or equal to √673 ≈ 25.9. Since 673 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 673 is a prime number.

Here’s another way we know that 673 is a prime number: Since 673÷4 has a remainder of one, and 23² + 12² = 673, and 23 and 12 have no common prime factors, 673 will be prime unless it is divisible by a primitive Pythagorean hypotenuse less than or equal to √673 ≈ 25.9. Since 673 is not divisible by 5, 13, or 17, we know that 673 is a prime number.

673 Logic

Don’t get discouraged if you make a mistake. Just fix it and move on. That’s what Ricardo did:

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664 The Back of a Halloween Cat’s Head

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664  in base 12 is the palindrome 474. Note that 4(144) + 7(12) + 4(1)= 664.

664 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-10-26

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  • 664 is a composite number.
  • Prime factorization: 664 = 2 x 2 x 2 x 83, which can be written 664 = (2^3) x 83
  • 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 664 has exactly 8 factors.
  • Factors of 664: 1, 2, 4, 8, 83, 166, 332, 664
  • Factor pairs: 664 = 1 x 664, 2 x 332, 4 x 166, or 8 x 83
  • Taking the factor pair with the largest square number factor, we get √664 = (√4)(√166) = 2√166 ≈ 25.768197.

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664 Logic

Ricardo tweeted the solution:

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655 and Level 6

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655  is the hypotenuse of the Pythagorean triple 393-524-655. What is the greatest common factor of those three numbers?

655 is a leg in exactly three triples. One of them is primitive; the rest are not. Which is which?

  • 655-1572-1703
  • 655-42900-42905
  • 655-214512-214513

Which of 655’s factors are the greatest common factors of each of the two that are not primitive?

655 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

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  • 655 is a composite number.
  • Prime factorization: 655 = 5 x 131
  • 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 655 has exactly 4 factors.
  • Factors of 655: 1, 5, 131, 655
  • Factor pairs: 655 = 1 x 655 or 5 x 131
  • 655 has no square factors that allow its square root to be simplified. √655 ≈ 25.592968.

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655 Logic

649 and Level 6

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6 – 4 + 9 = 11 so 649 is divisible by 11.

649 is the short leg in exactly three Pythagorean triples. Can you determine which one is a primitive triple, and what are the greatest common factors of each of the two non-primitive triples?

  • 649-3540-3599
  • 649-19140-19151
  • 649-210600-210601

649 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-10-12

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  • 649 is a composite number.
  • Prime factorization: 649 = 11 x 59
  • 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 649 has exactly 4 factors.
  • Factors of 649: 1, 11, 59, 649
  • Factor pairs: 649 = 1 x 649 or 11 x 59
  • 649 has no square factors that allow its square root to be simplified. √649 ≈ 25.475478.

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649 Logic

641 and Level 6

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25² + 4² = 641

641 is the hypotenuse of the primitive Pythagorean triple 200-609-641 which was calculated using 2(25)(4), 25² – 4², and 25² + 4² .

 641 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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  • 641 is a prime number.
  • Prime factorization: 641 is prime.
  • The exponent of prime number 641 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 641 has exactly 2 factors.
  • Factors of 641: 1, 641
  • Factor pairs: 641 = 1 x 641
  • 641 has no square factors that allow its square root to be simplified. √641 ≈ 25.3179778.

How do we know that 641 is a prime number? If 641 were not a prime number, then it would be divisible by at least one prime number less than or equal to √641 ≈ 25.3. Since 641 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 641 is a prime number.

Here’s another way we know that 641 is a prime number: Since 25²+ 4² = 641, an odd number, and 25 and 4 have no common prime factors, we know that 641 is a prime number simply because it is not divisible by 5, 13, or 17.

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641 Logic

634 and Level 6

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634 is the sum of the eighteen prime numbers from 5 to 71. Can you name them all?

Every natural number greater than 2 is part of at least one Pythagorean triple. Where can we find 634 in a Pythagorean triple?

  • 634 is the hypotenuse of only one Pythagorean triple: 150-616-634.
  • Also 634 is a leg in exactly one triple: 634-100488-100490.

The greatest common factor for either triple is 2.

634 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-28

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  • 634 is a composite number.
  • Prime factorization: 634 = 2 x 317
  • 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 634 has exactly 4 factors.
  • Factors of 634: 1, 2, 317, 634
  • Factor pairs: 634 = 1 x 634 or 2 x 317
  • 634 has no square factors that allow its square root to be simplified. √634 ≈ 25.1793566.

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634 Logic