Level 1 Puzzle

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

565 and Level 1

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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

558 and Level 1

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558 is the sum of consecutive prime numbers 277 and 281.

558  is also the sum of all the prime numbers from 7 to 67.

558 Puzzle

 

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

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  • 558 is a composite number.
  • Prime factorization: 558 = 2 x 3 x 3 x 31, which can be written 558 = 2 x (3^2) x 31
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 558 has exactly 12 factors.
  • Factors of 558: 1, 2, 3, 6, 9, 18, 31, 62, 93, 186, 279, 558
  • Factor pairs: 558 = 1 x 558, 2 x 279, 3 x 186, 6 x 93, 9 x 62, or 18 x 31
  • Taking the factor pair with the largest square number factor, we get √558 = (√9)(√62) = 3√62 ≈ 23.62202

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558 Factors

551 and Level 1

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551 is the sum of all the prime numbers from 11 to 67. It also equals this sum of consecutive primes: 179 + 181 + 191.

551 is the hypotenuse of the Pythagorean triple 380-399-551. What is the greatest common factor of those three numbers? Hint: It’s one of the factors of 551 listed below the puzzle.

551 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-07-13

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  • 551 is a composite number.
  • Prime factorization: 551 = 19 x 29
  • 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 551 has exactly 4 factors.
  • Factors of 551: 1, 19, 29, 551
  • Factor pairs: 551 = 1 x 551 or 19 x 29
  • 551 has no square factors that allow its square root to be simplified. √551 ≈ 23.453389

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551 Factors

544 and Level 1

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544 is the hypotenuse of Pythagorean triple 256-480-544. Can you find the greatest common factor of those three numbers?

544 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-07-06

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  • 544 is a composite number.
  • Prime factorization: 544 = 2 x 2 x 2 x 2 x 2 x 17, which can be written 544 = (2^5) x 17
  • The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 x 2 = 12. Therefore 544 has exactly 12 factors.
  • Factors of 544: 1, 2, 4, 8, 16, 17, 32, 34, 68, 136, 272, 544
  • Factor pairs: 544 = 1 x 544, 2 x 272, 4 x 136, 8 x 68, 16 x 34, or 17 x 32
  • Taking the factor pair with the largest square number factor, we get √544 = (√16)(√34) = 4√34 ≈ 23.3238

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544 Factors

 

537 and Level 1

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537 is made from 3 consecutive odd numbers so it is divisible by 3.

537 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-29

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  • 537 is a composite number.
  • Prime factorization: 537 = 3 x 179
  • 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 537 has exactly 4 factors.
  • Factors of 537: 1, 3, 179, 537
  • Factor pairs: 537 = 1 x 537 or 3 x 179
  • 537 has no square factors that allow its square root to be simplified. √537 ≈ 23.17326

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537 Factors

530 and Level 1

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The first three perfect numbers are 6, 28, and 496. Their sum is 530.

Can you find the greatest common factors for each of these Pythagorean triples?

  • 46-528-530
  • 192-494-530
  • 280-450-530
  • 318-424-530

530 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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  • 530 is a composite number.
  • Prime factorization: 530 = 2 x 5 x 53
  • 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 530 has exactly 8 factors.
  • Factors of 530: 1, 2, 5, 10, 53, 106, 265, 530
  • Factor pairs: 530 = 1 x 530, 2 x 265, 5 x 106, or 10 x 53
  • 530 has no square factors that allow its square root to be simplified. √530 ≈ 23.02172886644

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530 Factors

523 and Level 1

/ ivasallay / 2 Comments

523 = 61 + 67 + 71 + 73 + 79 + 83 + 89 which are all the prime numbers between 60 and 96.

521 and 523 are twin primes.

523 is the 99th prime number. It will take longer than ever before for another prime number to be featured on this blog.

The prime gap is defined as the difference between two consecutive prime numbers. The gap between this prime number and the next one is greater than between any two previous prime numbers.

GAP 100 Primes

Here is today’s puzzle:

523 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-15

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  • 523 is a prime number.
  • Prime factorization: 523 is prime.
  • The exponent of prime number 523 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 523 has exactly 2 factors.
  • Factors of 523: 1, 523
  • Factor pairs: 523 = 1 x 523
  • 523 has no square factors that allow its square root to be simplified. √523 ≈ 22.869193

How do we know that 523 is a prime number? If 523 were not a prime number, then it would be divisible by at least one prime number less than or equal to √523 ≈ 22.9. Since 523 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 523 is a prime number.

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523 Factors

516 Is an Untouchable Number

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Look at this chart of the sum of the factors for the numbers 1 – 25:

sum of factors

If we made the chart infinitely long using every counting number as n, there are certain numbers like 2, 5, 52, 88, and 96 that will NEVER appear in either column C or column D. Those numbers are called untouchable numbers, and 516 is one of them. Even though there are relatively few untouchable numbers, Paul Erdős proved that there are infinitely many of them.

516 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-08

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  • 516 is a composite number.
  • Prime factorization: 516 = 2 x 2 x 3 x 43, which can be written 516 = (2^2) x 3 x 43
  • 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 516 has exactly 12 factors.
  • Factors of 516: 1, 2, 3, 4, 6, 12, 43, 86, 129, 172, 258, 516
  • Factor pairs: 516 = 1 x 516, 2 x 258, 3 x 172, 4 x 129, 6 x 86, or 12 x 43
  • Taking the factor pair with the largest square number factor, we get √516 = (√4)(√129) = 2√129 ≈ 22.715633382

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516 Factors

509 and Level 1

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509 = 22^2 + 5^2.

509 is the hypotenuse of the primitive Pythagorean triple, 220-459-509.

509 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-06-01

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  • 509 is a prime number.
  • Prime factorization: 509 is prime.
  • The exponent of prime number 509 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 509 has exactly 2 factors.
  • Factors of 509: 1, 509
  • Factor pairs: 509 = 1 x 509
  • 509 has no square factors that allow its square root to be simplified. √509 ≈ 22.561028

How do we know that 509 is a prime number? If 509 were not a prime number, then it would be divisible by at least one prime number less than or equal to √509 ≈ 22.6. Since 509 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 509 is a prime number.

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509 Factors