Prime factorization

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

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

 

580 and Level 2

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580 is the sum of all the prime numbers from 83 to 107. Do you know what those consecutive prime numbers are?

580 is also the hypotenuse of four Pythagorean triples. In each case can you find the factor of 580 that is the greatest common factor of the triple?

  • 68-576-580
  • 96-572-580
  • 348-464-580
  • 400-420-580

580 Puzzle

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

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  • 580 is a composite number.
  • Prime factorization: 580 = 2 x 2 x 5 x 29, which can be written 580 = (2^2) x 5 x 29
  • 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 580 has exactly 12 factors.
  • Factors of 580: 1, 2, 4, 5, 10, 20, 29, 58, 116, 145, 290, 580
  • Factor pairs: 580 = 1 x 580, 2 x 290, 4 x 145, 5 x 116, 10 x 58, or 20 x 29
  • Taking the factor pair with the largest square number factor, we get √580 = (√4)(√145) = 2√145 ≈ 24.083189

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580 Factors

579 and Level 1

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579 is the hypotenuse of the Pythagorean triple 285-504-579. Which of 579’s factors is the greatest common factor of those three numbers?

Last week someone googled “find,the least 6 digit which has 173 factor” and arrived at Findthefactors.com. Here how to find the answer: Divide 100,000 by 173 and get 578.03 approximately. Round that answer up to 579. Multiply 579 by 173 and get 100167, the smallest 6-digit number that has 173 as a factor.

579 Puzzle

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

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  • 579 is a composite number.
  • Prime factorization: 579 = 3 x 193
  • 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 579 has exactly 4 factors.
  • Factors of 579: 1, 3, 193, 579
  • Factor pairs: 579 = 1 x 579 or 3 x 193
  • 579 has no square factors that allow its square root to be simplified. √579 ≈ 24.0624188

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579 Factors

576 and Level 5

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The number formed from the last 2 digits of 576 is divisible by 4. That means 576 is also divisible by 4.

576 is made from 3 consecutive numbers so it is divisible by 3. Since the middle number, 6, is also divisible by 3, we know that 576 is also divisible by 9.

Either of the previous statements is enough to indicate that √576 can be reduced. In fact, boy, can it ever be reduced!

576 is the smallest number to have exactly 21 factors. (Only perfect squares have an odd number of factors.)

576 is also the sum of consecutive primes in two different ways:

  • 283 + 293 = 576
  • 137 + 139 + 149 + 151 = 576

576 Puzzle

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

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  • 576 is a composite number.
  • Prime factorization: 576 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3, which can be written 576 = (2^6) x (3^2)
  • The exponents in the prime factorization are 6 and 2. Adding one to each and multiplying we get (6 + 1)(2 + 1) = 7 x 3 = 21. Therefore 576 has exactly 21 factors.
  • Factors of 576: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 576
  • Factor pairs: 576 = 1 x 576, 2 x 288, 3 x 192, 4 x 144, 6 x 96, 8 x 72, 9 x 64, 12 x 48, 16 x 36, 18 x 32 or 24 x 24
  • 576 is a perfect square. √576 = 24

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576 Logic

575 and Level 4

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575 is the hypotenuse of two Pythagorean triples: 345-460-575 and 161-552-575. What is the greatest common factor of each of those triples?

575 = 23 x 25, which can be written (24 – 1)(24 + 1) or (24^2) – 1.

575 Puzzle

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

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  • 575 is a composite number.
  • Prime factorization: 575 = 5 x 5 x 23, which can be written 575 = (5^2) x 23
  • 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 575 has exactly 6 factors.
  • Factors of 575: 1, 5, 23, 25, 115, 575
  • Factor pairs: 575 = 1 x 575, 5 x 115, or 23 x 25
  • Taking the factor pair with the largest square number factor, we get √575 = (√25)(√23) = 3√23 ≈ 23.9791576

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575 Logic

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