Month: August 2015

590 and Level 5

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3(20^2)/2 – 20/2 = 600 – 10 = 590. That means that 590 is the 20th pentagonal number.

590 is also the hypotenuse of the Pythagorean triple 354-472-590. Which factor of 590 is the greatest common factor of those three numbers? Hint: In this case the GCF is the difference between the middle number in the triple and either of the other two numbers.

590 Puzzle

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

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  • 590 is a composite number.
  • Prime factorization: 590 = 2 x 5 x 59
  • 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 590 has exactly 8 factors.
  • Factors of 590: 1, 2, 5, 10, 59, 118, 295, 590
  • Factor pairs: 590 = 1 x 590, 2 x 295, 5 x 118, or 10 x 59
  • 590 has no square factors that allow its square root to be simplified. √590 ≈ 24.2899

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590 Logic

589 and Level 4

/ ivasallay / 2 Comments

589 is the sum of the three prime numbers from 193 to 199. It should be pretty obvious what that middle prime number is.

589 Puzzle

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

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  • 589 is a composite number.
  • Prime factorization: 589 = 19 x 31
  • 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 589 has exactly 4 factors.
  • Factors of 589: 1, 19, 31, 589
  • Factor pairs: 589 = 1 x 589 or 19 x 31
  • 589 has no square factors that allow its square root to be simplified. √589 ≈ 24.269322

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589 Logic

Simplifying √588 and Level 3

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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

587 and Level 2

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587 is the sum of the five prime numbers from 107 to 131.

587 Puzzle

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

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  • 587 is a prime number.
  • Prime factorization: 587 is prime.
  • The exponent of prime number 587 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 587 has exactly 2 factors.
  • Factors of 587: 1, 587
  • Factor pairs: 587 = 1 x 587
  • 587 has no square factors that allow its square root to be simplified. √587 ≈ 24.22808

How do we know that 587 is a prime number? If 587 were not a prime number, then it would be divisible by at least one prime number less than or equal to √587 ≈ 24.2. Since 587 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 587 is a prime number.

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587 Factors

586 and Level 1

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586 is the hypotenuse of the Pythagorean triple 136-570-586. Which factor of 586 is the greatest common factor of those three numbers?

586 Puzzle

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

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  • 586 is a composite number.
  • Prime factorization: 586 = 2 x 293
  • 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 586 has exactly 4 factors.
  • Factors of 586: 1, 2, 293, 586
  • Factor pairs: 586 = 1 x 586 or 2 x 293
  • 586 has no square factors that allow its square root to be simplified. √586 ≈ 24.2074

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586 Factors

What can you see in the number 585?

/ ivasallay / 1 Comment

This week I watched an excellent video titled 5 x 9 is more than 45. Indeed 45 is so much more than simply 5 x 9. Every multiplication fact is much more than that mere fact, but Steve Wyborney used 5 x 9 = 45 in his video… Guess what! 585 is a multiple of 45.

As I thought about the number 585, I marveled at some of the hidden mysteries this number holds.

Since 585 is divisible by two different centered square numbers, 5 and 13, I saw that 585 could be represented by this lovely array that has 45 larger squares made up of 13 smaller colorful squares. When you look at the array, do you just see 585 squares or can you see even more multiplication and division facts? If you rotate the array 90 degrees, do the facts change?

585 Squares-1

What do you see in this array of 117 medium sized squares made up of 5 smaller squares:

585 Squares-2

Or this more simple looking array of sixty-five 3 x 3 squares.

585 Squares-3

All of these arrays are in just two dimensions. A 5 x 9 x 13 rectangular prism is ONE way to represent 585 in three dimensions.

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Since factors 5, 13, and 65 are hypotenuses of primitive Pythagorean triples, 585 is the hypotenuse of four Pythagorean triples. Each triple has a different greatest common factor. Can you figure out what each one is?

  • 144-567-585
  • 225-540-585
  • 297-504-585
  • 351-468-585

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And if all of that wasn’t enough, OEIS.org informs us that 585 is a palindrome in 3 different bases!

  • 585 = 1001001001 in base 2
  • 585 = 1111 in base 8
  • 585 = 585 in base 10

Here are two different methods of determining 585 in base 8. In both methods the base 8 representation is in blue.

585 base 8

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  • 585 is a composite number.
  • Prime factorization: 585 = 3 x 3 x 5 x 13, which can be written 585 = (3^2) x 5 x 13
  • 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 585 has exactly 12 factors.
  • Factors of 585: 1, 3, 5, 9, 13, 15, 39, 45, 65, 117, 195, 585
  • Factor pairs: 585 = 1 x 585, 3 x 195, 5 x 117, 9 x 65, 13 x 45, or 15 x 39
  • Taking the factor pair with the largest square number factor, we get √585 = (√9)(√65) = 3√65 ≈ 24.18677

584 and Level 6

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584 is the hypotenuse of the Pythagorean triple 384-440-584. Which factor of 584 is the greatest common factor of those three numbers?

584 Puzzle

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

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  • 584 is a composite number.
  • Prime factorization: 584 = 2 x 2 x 2 x 73, which can be written 584 = (2^3) x 73
  • 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 584 has exactly 8 factors.
  • Factors of 584: 1, 2, 4, 8, 73, 146, 292, 584
  • Factor pairs: 584 = 1 x 584, 2 x 292, 4 x 146, or 8 x 73
  • Taking the factor pair with the largest square number factor, we get √584 = (√4)(√146) = 2√146 ≈ 24.16609

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584 Logic

583 The secret to solving these puzzles is using logic and never guessing

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5 – 8 + 3 = 0. Since zero is divisible by 11 we know that 583 is also divisible by 11.

583 is the sum of all the prime numbers from 31 to 73. Add them up and see for yourself.

583 is also the hypotenuse of the Pythagorean triple 308-495-583. What is the greatest common factor of those three numbers?

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The secret to solving these puzzles is using logic and never guessing. That means NEVER write a factor in a cell unless you know for sure it belongs there. Not following this advice can cause frustration when the puzzles are only intended to be fun. Read How to Solve a FIND THE FACTORS puzzle for more instructions.

583 Puzzle

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  • 583 is a composite number.
  • Prime factorization: 583 = 11 x 53
  • 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 583 has exactly 4 factors.
  • Factors of 583: 1, 11, 53, 583
  • Factor pairs: 583 = 1 x 583 or 11 x 53
  • 583 has no square factors that allow its square root to be simplified. √583 ≈ 24.14539

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583 Logic

582 and Level 4

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582 is the sum of all the prime numbers from 59 to 89. Can you list those eight consecutive primes?

582 is also the hypotenuse of the Pythagorean triple 390-432-582. Which of 582’s factors is the greatest common factor of those three numbers?

582 Puzzle

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

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  • 582 is a composite number.
  • Prime factorization: 582 = 2 x 3 x 97
  • 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 582 has exactly 8 factors.
  • Factors of 582: 1, 2, 3, 6, 97, 194, 291, 582
  • Factor pairs: 582 = 1 x 582, 2 x 291, 3 x 194, or 6 x 97
  • 582 has no square factors that allow its square root to be simplified. √582 ≈ 24.124676

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582 Logic

581 and Level 3

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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