Level 5 Puzzle

606 and Level 5

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606 is the sum of the six prime numbers from 89 to 109

606 is also the hypotenuse of the Pythagorean triple 120-594-606. What is the greatest common factor of those three numbers?

606 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-01

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  • 606 is a composite number.
  • Prime factorization: 606 = 2 x 3 x 101
  • 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 606 has exactly 8 factors.
  • Factors of 606: 1, 2, 3, 6, 101, 202, 303, 606
  • Factor pairs: 606 = 1 x 606, 2 x 303, 3 x 202, or 6 x 101
  • 606 has no square factors that allow its square root to be simplified. √606 ≈ 24.617067

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597 and Level 5

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

Because it is divisible by 3, it is the sum of three consecutive numbers. It is also the sum of three consecutive odd numbers:

  • 198 + 199 + 200 = 597
  • 197 + 199 + 201 = 597

Why was 199 the middle number in both of those sums?

OEIS.org informs us that even though (597!!! + 1) generates an error message on our calculators, it is indeed a very large prime number. Whoever discovered that fact must have been extremely excited.

597 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-24

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  • 597 is a composite number.
  • Prime factorization: 597 = 3 x 199
  • 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 597 has exactly 4 factors.
  • Factors of 597: 1, 3, 199, 597
  • Factor pairs: 597 = 1 x 597 or 3 x 199
  • 597 has no square factors that allow its square root to be simplified. √597 ≈ 24.43358

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597 Logic

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

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

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

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

563 and Level 5

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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.
  • 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

548 and Level 5

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548 is the hypotenuse of the Pythagorean triple 352-420-548. What is the greatest common factor of those three numbers?

548 Puzzle

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

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  • 548 is a composite number.
  • Prime factorization: 548 = 2 x 2 x 137, which can be written 548 = (2^2) x 137
  • 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 548 has exactly 6 factors.
  • Factors of 548: 1, 2, 4, 137, 274, 548
  • Factor pairs: 548 = 1 x 548, 2 x 274, or 4 x 137
  • Taking the factor pair with the largest square number factor, we get √548 = (√4)(√137) = 2√137 ≈ 23.4093998

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548 Logic

541 and Level 5

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541 = 21² + 10² = 441 + 100

541 is the hypotenuse of the primitive Pythagorean triple 341-420-541

And finally, after the longest string of composite numbers so far, 541 is quite notably a prime number. In fact, it is the 100th prime number.

541 Puzzle

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

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  • 541 is a prime number.
  • Prime factorization: 541 is prime.
  • The exponent of prime number 541 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 541 has exactly 2 factors.
  • Factors of 541: 1, 541
  • Factor pairs: 541 = 1 x 541
  • 541 has no square factors that allow its square root to be simplified. √541 ≈ 23.25940 66 99 22 601 44

How do we know that 541 is a prime number? If 541 were not a prime number, then it would be divisible by at least one prime number less than or equal to √541 ≈ 23.3. Since 541 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 541 is a prime number.

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541 Logic

527 and Level 5

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527 is the sum of consecutive primes including all the prime numbers in its prime factorization. Can you list those consecutive primes?

527 Puzzle

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

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  • 527 is a composite number.
  • Prime factorization: 527 = 17 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 527 has exactly 4 factors.
  • Factors of 527: 1, 17, 31, 527
  • Factor pairs: 527 = 1 x 527 or 17 x 31
  • 527 has no square factors that allow its square root to be simplified. √527 ≈ 22.95648

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527 Logic