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

573 and Level 2

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573 is made from three consecutive odd numbers so it is divisible by 3. The number in the middle, 5, is not divisible by 3, so 573 is not divisible by 9.

573 squared is 328,329, a 6 digit number that looks like two consecutive numbers! Thank you OEIS.org for that interesting fact.

573 Puzzle

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

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  • 573 is a composite number.
  • Prime factorization: 573 = 3 x 191
  • 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 573 has exactly 4 factors.
  • Factors of 573: 1, 3, 191, 573
  • Factor pairs: 573 = 1 x 573 or 3 x 191
  • 573 has no square factors that allow its square root to be simplified. √573 ≈ 23.9374

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573 Factors

572 and Level 1

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72 can be evenly divided by 4 so 572 is divisible by 4.

5 – 7 + 2 = 0, which can be evenly divided by 11, so 11 is a factor of 572.

572 is the hypotenuse of the Pythagorean triple 220-528-572. Which factor of 572 is the greatest common factor of those three numbers?

572 Puzzle

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

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  • 572 is a composite number.
  • Prime factorization: 572 = 2 x 2 x 11 x 13, which can be written 572 = (2^2) x 11 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 572 has exactly 12 factors.
  • Factors of 572: 1, 2, 4, 11, 13, 22, 26, 44, 52, 143, 286, 572
  • Factor pairs: 572 = 1 x 572, 2 x 286, 4 x 143, 11 x 52, 13 x 44, or 22 x 26
  • Taking the factor pair with the largest square number factor, we get √572 = (√4)(√143) = 2√143 ≈ 23.91652

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572 Factors

571 Family Time

/ ivasallay / 5 Comments

For the last two weeks we’ve been spending time with our grandchildren in Salt Lake City, Utah; Portland, Oregon; Houston, Texas; and Hartford, Connecticut. We also spent time with my brother’s daughter and her two children. AND we celebrated my sister’s 70th birthday party and reconnected with all of her children in Portland. We have been busy, and I am tired, but happy.

We’ve had a wonderful time with all of them, and it was so painful to say goodbye. We and a few other people took pictures except we didn’t get any pictures of the grandkids that live within 10 miles of our home. What’s with that? I’m as guilty as anyone else who takes people and things for granted.

Here’s a picture of me holding my newest grandson. My daughter is in the background holding one of her nieces.

 

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Since 171 + 190 + 210 = 571, it is the sum of the 18th, 19th, and 20th triangular numbers. That makes 571 the 20th centered triangular number.

  • 571 is a prime number and a twin prime with 569.
  • Prime factorization: 571 is prime.
  • The exponent of prime number 571 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 571 has exactly 2 factors.
  • Factors of 571: 1, 571
  • Factor pairs: 571 = 1 x 571
  • 571 has no square factors that allow its square root to be simplified. √571 ≈ 23.8956

How do we know that 571 is a prime number? If 571 were not a prime number, then it would be divisible by at least one prime number less than or equal to √571 ≈ 23.9. Since 571 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 571 is a prime number.

 

 

570 and Level 6

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570 is the hypotenuse of Pythagorean triple 342-456-570. What 3 digit number is the greatest common factor of those three numbers?

570 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-07-27

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  • 570 is a composite number.
  • Prime factorization: 570 = 2 x 3 x 5 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 570 has exactly 16 factors.
  • Factors of 570: 1, 2, 3, 5, 6, 10, 15, 19, 30, 38, 57, 95, 114, 190, 285, 570
  • Factor pairs: 570 = 1 x 570, 2 x 285, 3 x 190, 5 x 114, 6 x 95, 10 x 57, 15 x 38, or 19 x 30
  • 570 has no square factors that allow its square root to be simplified. √570 ≈ 23.87467

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570 Logic

569 and Level 5

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When we divide the last two digits of prime number 569 by 4, we get a remainder of one. That means that 569 is the sum of two square numbers, specifically, 20² + 13² = 569.

From those two square numbers we will obtain 569 as the hypotenuse of the primitive Pythagorean triple 231-520-569:

  • 20² – 13² = 231
  • 2 x 13 x 20 = 520
  • 20² + 13² = 569

569 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-07-27

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  • 569 is a prime number.
  • Prime factorization: 569 is prime.
  • The exponent of prime number 569 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 569 has exactly 2 factors.
  • Factors of 569: 1, 569
  • Factor pairs: 569 = 1 x 569
  • 569 has no square factors that allow its square root to be simplified. √569 ≈ 23.85372

How do we know that 569 is a prime number? If 569 were not a prime number, then it would be divisible by at least one prime number less than or equal to √569 ≈ 23.8. Since 569 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 569 is a prime number.

569 Logic

568 and Level 4

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568 is the sum of the first 19 prime numbers (all the primes from 2 to 67).

568 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-07-27

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  • 568 is a composite number.
  • Prime factorization: 568 = 2 x 2 x 2 x 71, which can be written 568 = (2^3) x 71
  • 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 568 has exactly 8 factors.
  • Factors of 568: 1, 2, 4, 8, 71, 142, 284, 568
  • Factor pairs: 568 = 1 x 568, 2 x 284, 4 x 142, or 8 x 71
  • Taking the factor pair with the largest square number factor, we get √568 = (√4)(√142) = 2√142 ≈ 23.83275

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568 Logic

567 and Level 3

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567 is made from three consecutive numbers so it is divisible by 3. Since the middle number of those three consecutive numbers is divisible by 3, we know that 567 is also divisible by 9.

567 and its square, 321489, use all the digits 1-9 exactly once. I learned that fact from reading OEIS.org.

567 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-07-27

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  • 567 is a composite number.
  • Prime factorization: 567 = 3 x 3 x 3 x 3 x 7, which can be written 567 = (3^4) x 7
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 x 2 = 10. Therefore 567 has exactly 10 factors.
  • Factors of 567: 1, 3, 7, 9, 21, 27, 63, 81, 189, 567
  • Factor pairs: 567 = 1 x 567, 3 x 189, 7 x 81, 9 x 63, or 21 x 27
  • Taking the factor pair with the largest square number factor, we get √567 = (√81)(√7) = 9√7 ≈ 23.81176

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567 Factors

 

566 and Level 2

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566 is the sum of all the prime numbers from 3 to 67.

566 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-07-27

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  • 566 is a composite number.
  • Prime factorization: 566 = 2 x 283
  • 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 566 has exactly 4 factors.
  • Factors of 566: 1, 2, 283, 566
  • Factor pairs: 566 = 1 x 566 or 2 x 283
  • 566 has no square factors that allow its square root to be simplified. √566 ≈ 23.79075

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566 Factors