Level 3 Puzzle

638 and Level 3

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638 is the sum of the four prime numbers from 151 to 167.

6 – 3 + 8 = 11. Thus 638 is divisible by 11.

638 is also the hypotenuse of the Pythagorean triple 440-462-638. What is the greatest common factor of those three numbers?

638 Puzzle

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

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  • 638 is a composite number.
  • Prime factorization: 638 = 2 x 11 x 29
  • 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 638 has exactly 8 factors.
  • Factors of 638: 1, 2, 11, 22, 29, 58, 319, 638
  • Factor pairs: 638 = 1 x 638, 2 x 319, 11 x 58, or 22 x 29
  • 638 has no square factors that allow its square root to be simplified. √638 ≈ 25.25866.

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 12.)  Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

638 Factors

631 and Level 3

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631 is a prime number that happens to also be the sum of the eleven prime numbers from 37 to 79.

Sometimes mathematicians talk about triangular numbers, centered triangular numbers, hexagonal numbers, and centered hexagonal numbers. Is there any relationship between those terms?

  • The sum of any three consecutive triangular numbers is a centered triangular number.
  • ALL hexagonal numbers are triangular numbers.
  • Most centered hexagonal numbers are NOT centered triangular numbers.

In my last post, I pointed out that 630 is a triangular number and a hexagonal number.

Well, amazingly the next number, 631, is a centered triangular number AND a centered hexagonal number.

How often does one more than a triangular number equal a centered triangular number? Also how often does one more than a hexagonal number equal a centered hexagonal number?

Let’s compare a couple of lists of these types of numbers up to 631. Red means a number is in both lists. Blue means the number in the 1st list is one less than the number in the 2nd list.

  • Triangular numbers (with hexagonal numbers in bold type):
  • 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630.
  • Centered triangular numbers (with numbers that are also centered hexagonal numbers in bold type):
  • 1, 4, 10, 19, 31, 46, 64, 85, 109, 136, 166, 199, 235, 274, 316, 361, 409, 460, 514, 571, 631.

1 is the only number in the lists that is a triangular, centered triangular, hexagonal, and centered hexagonal number all at the same time. (1 is also a square number, a pentagonal number, etc.)

(630, 631) is the first combination of numbers that has BOTH a triangular number immediately followed by a centered triangular number AND a hexagonal number immediately followed by a centered hexagonal number.

631 Puzzle

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

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  • 631 is a prime number.
  • Prime factorization: 631 is prime.
  • The exponent of prime number 631 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 631 has exactly 2 factors.
  • Factors of 631: 1, 631
  • Factor pairs: 631 = 1 x 631
  • 631 has no square factors that allow its square root to be simplified. √631 ≈ 25.11971.

How do we know that 631 is a prime number? If 631 were not a prime number, then it would be divisible by at least one prime number less than or equal to √631 ≈ 25.1. Since 631 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 631 is a prime number.

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 10.)  Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

631 Factors

624 and Level 3

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Since 624 is one less than 625 which is 25 squared, it should be obvious that 24 x 26 = 624.

624 is the hypotenuse of the Pythagorean triple 240-576-624. What is the greatest common factor of those three numbers?

624 is the sum of consecutive odd numbers and twin primes 311 and 313.

624 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-21

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  • 624 is a composite number.
  • Prime factorization: 624 = 2 x 2 x 2 x 2 x 3 x 13, which can be written 624 = (2^4) x 3 x 13
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 624 has exactly 20 factors.
  • Factors of 624: 1, 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 39, 48, 52, 78, 104, 156, 208, 312, 624
  • Factor pairs: 624 = 1 x 624, 2 x 312, 3 x 208, 4 x 156, 6 x 104, 8 x 78, 12 x 52, 13 x 48, 16 x 39 or 24 x 26
  • Taking the factor pair with the largest square number factor, we get √624 = (√16)(√39) = 4√39 ≈ 24.97999199.

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 12.)  Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

624 Factors

 

618 and Level 3

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618 is even so it is obviously divisible by 2. Is it divisible by 3?

Yes, 618 has the same digits as 16 & 8. Since 16 is 8 doubled, 618 is divisible by 3.

618  is the sum of consecutive prime numbers 307 and 311.

618 can be written as the sum of 4 consecutive numbers because 618 is greater than 6 and is divisible by 2, but not by 4. Thus 153 + 154 + 155 + 156 = 618.

618 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-14

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  • 618 is a composite number.
  • Prime factorization: 618 = 2 x 3 x 103
  • 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 618 has exactly 8 factors.
  • Factors of 618: 1, 2, 3, 6, 103, 206, 309, 618
  • Factor pairs: 618 = 1 x 618, 2 x 309, 3 x 206, or 6 x 103
  • 618 has no square factors that allow its square root to be simplified. √618 ≈ 24.8596.

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 10.)  Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

618 Factors

610 and Level 3

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The first 17 Fibonacci numbers are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597. Notice that 610 is the 15th Fibonacci number.

There is a fascinating relationship between some of the Fibonacci numbers and some of the Markov numbers. 610 is the 12th Markov number. Get out your calculator and satisfy yourself that the following two Diophantine equations involving Fibonacci/Markov numbers are true:

1² + 233² + 610² = 3(1)(233)(610)

1² + 610² + 1597² = 3(1)(610)(1597)

Here is a fascinating fact I learned from twitter:

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Starting with 5, every other Fibonacci number would be 5, 13, 34, 89, 233, 610, 1597, . . .

610 is on that list. What could be the integer sides of a right triangle with 610 as the hypotenuse?

There are actually FOUR such triangles, namely. . .

  • 110-600-610
  • 272-546-610
  • 414-448-610
  • 366-488-610

None of those are primitives, but it is a great list nonetheless!

610 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-09-07

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  • 610 is a composite number.
  • Prime factorization: 610 = 2 x 5 x 61
  • 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 610 has exactly 8 factors.
  • Factors of 610: 1, 2, 5, 10, 61, 122, 305, 610
  • Factor pairs: 610 = 1 x 610, 2 x 305, 5 x 122, or 10 x 61
  • 610 has no square factors that allow its square root to be simplified. √610 ≈ 24.698178.

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

610 Factors

604 and Level 3

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604 is not a power of 2, can be evenly divided by 4, but not by 8, and 604 is greater than 28. Those three facts together mean that 604 can be written as the sum of 8 consecutive whole numbers. To find the 8 numbers first divide 604 by 8 to get 75 1/2, then add the 4 counting numbers before 75 1/2 to the 4 counting numbers after it:

72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 = 604

604 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-01

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  • 604 is a composite number.
  • Prime factorization: 604 = 2 x 2 x 151, which can be written 604 = (2^2) x 151
  • 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 604 has exactly 6 factors.
  • Factors of 604: 1, 2, 4, 151, 302, 604
  • Factor pairs: 604 = 1 x 604, 2 x 302, or 4 x 151
  • Taking the factor pair with the largest square number factor, we get √604 = (√4)(√151) = 2√151 ≈ 24.576411

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

604 Factors

595 and Level 3

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595 = (34/2)(34 + 1) so 595 is the 34th triangular number.

Thus 1 + 2 + 3 + 4 + . . . . . + 31 + 32 + 33 + 34 = 595

595 is also the hypotenuse of four Pythagorean triples. Each of these triples has its own greatest common factor.

  • 91-588-595
  • 252-539-595
  • 280-525-595
  • 357-476-595

595 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-24

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  • 595 is a composite number.
  • Prime factorization: 595 = 5 x 7 x 17
  • 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 595 has exactly 8 factors.
  • Factors of 595: 1, 5, 7, 17, 35, 85, 119, 595
  • Factor pairs: 595 = 1 x 595, 5 x 119, 7 x 85, or 17 x 35
  • 595 has no square factors that allow its square root to be simplified. √595 ≈ 24.39262.

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

595 Factors

 

Simplifying √588 and Level 3

/ ivasallay / Leave a comment

88 is divisible by 4 so 588 is also divisible by 4, and that means √588 can be reduced.

About 83% of the numbers that have reducible square roots are divisible by 4 and/or by 9, and it is so easy to tell if even a very long number is divisible by either of those numbers. It is also easier to divide a number by 4 or 9 than it is to divide by their square roots twice.

When I reduce a square root, I like to make a little cake and start by dividing by 100, 4, or 9 if any of those numbers are its factors. Here are the steps I used to make a cake for 588 with as many perfect squares on the outside of the cake as possible.

  1. 588 ÷ 4 = 147
  2. 147 is not divisible by 4 again, but 5 + 8 + 8 = 21 so 147 is divisible by 3, but not by 9.
  3. 147 ÷ 3 = 49 which is a perfect square, so I stop dividing and simply take the square roots of everything on the outside of the cake and multiply them together.

This is what my cake looks like:

588 cake

And now for today’s Level 3 puzzle:

588 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-17

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  • 588 is a composite number.
  • Prime factorization: 588 = 2 x 2 x 3 x 7 x 7, which can be written 588 = (2^2) x 3 x (7^2)
  • The exponents in the prime factorization are 2, 1 and 2. Adding one to each and multiplying we get (2 + 1)(1 + 1)(2 + 1) = 3 x 2 x 3 = 18. Therefore 588 has exactly 18 factors.
  • Factors of 588: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 49, 84, 98, 147, 196, 294, 588
  • Factor pairs: 588 = 1 x 588, 2 x 294, 3 x 196, 4 x 147, 6 x 98, 7 x 84, 12 x 49, 14 x 42 or 21 x 28
  • Taking the factor pair with the largest square number factor, we get √588 = (√196)(√3) = 14√3 ≈ 24.248711

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

588 Factors

581 and Level 3

/ ivasallay / Leave a comment

A year ago I wrote a short, but heartfelt post for World Elephant Day. I invite you to read it or any other article that makes you aware of what you can do to protect elephants from poachers.

581 is the sum of consecutive primes two different ways: 191 + 193 + 197 = 581, and the sum of all the prime numbers from 19 to 71 also is 581.

581 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-10

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  • 581 is a composite number.
  • Prime factorization: 581 = 7 x 83
  • 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 581 has exactly 4 factors.
  • Factors of 581: 1, 7, 83, 581
  • Factor pairs: 581 = 1 x 581 or 7 x 83
  • 581 has no square factors that allow its square root to be simplified. √581 ≈ 24.10394

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

581 Factors

 

574 Start at the top of the first column and fly down one cell at a time to solve this Level 3 puzzle

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One of 574’s factors is the hypotenuse of a primitive Pythagorean triple so 574 is the hypotenuse of the triple 126-560-574. Which of 574’s factors is the greatest common factor of those three numbers?

574 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-03

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  • 574 is a composite number.
  • Prime factorization: 574 = 2 x 7 x 41
  • 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 574 has exactly 8 factors.
  • Factors of 574: 1, 2, 7, 14, 41, 82, 287, 574
  • Factor pairs: 574 = 1 x 574, 2 x 287, 7 x 82, or 14 x 41
  • 574 has no square factors that allow its square root to be simplified. √574 ≈ 23.958297.

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A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

574 Factors