A Multiplication Based Logic Puzzle

Archive for July, 2015

569 and Level 5

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, (20^2) + (13^2) = 569.

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

  • (20^2) – (13^2) = 231
  • 2 x 13 x 20 = 520
  • (20^2) + (13^2) = 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 and cannot be factored.
  • 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

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568 and Level 4

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

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 stetson.edu.

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

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

 

565 and Level 1

565 is the sum of consecutive primes: 181 + 191 + 193 = 565.

565 is the sum of two squares two different ways: 565 = (22^2) + (9^2), and 565 = (23^2) + (6^2).

565 is the hypotenuse of four Pythagorean triples. The greatest common factor of two of them is 1 because they are primitives. Which of these triples are not primitive, and what is the greatest common factor of each of them?

  • 75-560-565
  • 276-493-565
  • 339-452-565
  • 396-403-565

565 Puzzle

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

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  • 565 is a composite number.
  • Prime factorization: 565 = 5 x 113
  • 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 565 has exactly 4 factors.
  • Factors of 565: 1, 5, 113, 565
  • Factor pairs: 565 = 1 x 565 or 5 x 113
  • 565 has no square factors that allow its square root to be simplified. √565 ≈ 23.7697.

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565 Factors

564 and Level 6

Consecutive primes 281 and 283 add up to 564.

564 is made from three consecutive numbers so it can be evenly divided by 3. If the middle number is divisible by 3, then a number made from three consecutive numbers will also be divisible by 9. Is 564 divisible by 9? Why or why not?

564 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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  • 564 is a composite number.
  • Prime factorization: 564 = 2 x 2 x 3 x 47, which can be written 564 = (2^2) x 3 x 47
  • 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 564 has exactly 12 factors.
  • Factors of 564: 1, 2, 3, 4, 6, 12, 47, 94, 141, 188, 282, 564
  • Factor pairs: 564 = 1 x 564, 2 x 282, 3 x 188, 4 x 141, 6 x 94, or 12 x 47
  • Taking the factor pair with the largest square number factor, we get √564 = (√4)(√141) = 2√141 ≈ 23.74868

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564 Logic

563 and Level 5

563 is prime, and it is the sum of all the prime numbers from 5 to 67.

563 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-20

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  • 563 is a prime number.
  • Prime factorization: 563 is prime and cannot be factored.
  • The exponent of prime number 563 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 563 has exactly 2 factors.
  • Factors of 563: 1, 563
  • Factor pairs: 563 = 1 x 563
  • 563 has no square factors that allow its square root to be simplified. √563 ≈ 23.727621

How do we know that 563 is a prime number? If 563 were not a prime number, then it would be divisible by at least one prime number less than or equal to √563 ≈ 23.7. Since 563 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 563 is a prime number.

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563 Logic

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