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, 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

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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.
  • 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

562 and Level 4

562 is the sum of all the prime numbers from 23 to 71.

562 is also the hypotenuse of the Pythagorean triple 320-462-562. Which factor of 562 is the greatest common factor of those three numbers?

562 Puzzle

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

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  • 562 is a composite number.
  • Prime factorization: 562 = 2 x 281
  • 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 562 has exactly 4 factors.
  • Factors of 562: 1, 2, 281, 562
  • Factor pairs: 562 = 1 x 562 or 2 x 281
  • 562 has no square factors that allow its square root to be simplified. √562 ≈ 23.7065

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562 Logic

561 gives a false positive to these 102 prime number tests

We can use some easy divisibility tests to find two of the factors of 561.

  • 5 + 6 + 1 = 12, a multiple of 3 so 561 is divisible by 3.
  • 5 – 6 + 1 = 0, which is divisible by 11 so 561 can be evenly divided by 11.

But if we only do those divisibility tests, we will miss something very significant about the number 561:

If you divide 2^561 by 561, the remainder will be 2. When we are more interested in the remainder than the quotient, we can simply type “2, x^y, 561, Mod, 561, =” into the computer’s scientific calculator:

561 Mod Calculator

This is only a picture of a calculator.

2^561 (mod 561) = 2 means that 561 is VERY LIKELY a prime number, but this is one time when VERY LIKELY does not mean ACTUALLY!

561 has something in common with the number 341. Yes, both of them pass this quick prime number test, and both of them are composite numbers divisible by 11. Both numbers are called pseudo-prime numbers. (341 and 561 are the two smallest composite numbers to give a false positive to this particular test.)

561 is even more remarkable than 341:

  • 2^561 (mod 561) = 2, and 2^341 (mod 341) = 2 (Both numbers pass.)
  • 3^561 (mod 561) = 3, while 3^341 (mod 341) = 168 (561 passes; 341 fails.)
  • 5^561 (mod 561) = 5
  • 7^561 (mod 561) = 7
  • 11^561 (mod 561) = 11
  • 13^561 (mod 561) = 13
  • 17^561 (mod 561) = 17
  • etc.

There are 102 prime numbers less than 561, and p^561 (mod 561) = p for every single one of them! 561 acts like a prime number in those 102 ways.

In 1910 R. D. Carmichael discovered that 561 is the first COMPOSITE number that passes ALL those modular (remainder) prime number tests, so 561 is the first Carmichael number. Yes, there will be more – in fact, infinitely more.

R. D. Carmichael actually found that 561 passes ALL 559 prime number tests using each whole number between 1 and 561, for example 33^561 (mod 561) = 33. All prime numbers can make a similar claim, but 561 is the smallest composite number with that property.

(Note: I did not use the standard mathematical notation for this property, but what I used is equivalent to it and doesn’t require parenthesis when typing it into the computer’s scientific calculator. Also I think “=” is less intimidating looking for some of my readers than “≡”.)

There are other reasons why the number 561 is an interesting number:

Because 33 x 34/2 = 561, we know that 561 is the 33rd triangular number and is equal to 1 + 2 + 3 + . . . + 31 + 32 + 33, the sum of the first 33 whole numbers.

Because 17 x (2 x 17 – 1) = 561, we know that 561 is the 17th hexagonal number. (All hexagonal numbers are also triangular numbers.)

561 is also the hypotenuse of the Pythagorean triple 264-495-561. What is the greatest common factor of those three numbers? Hint: it is one of the factors of 561 listed below:

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  • 561 is a composite number.
  • Prime factorization: 561 = 3 x 11 x 17
  • 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 561 has exactly 8 factors.
  • Factors of 561: 1, 3, 11, 17, 33, 51, 187, 561
  • Factor pairs: 561 = 1 x 561, 3 x 187, 11 x 51, or 17 x 33
  • 561 has no square factors that allow its square root to be simplified. √561 ≈ 23.6854

560 and Level 3

560 is the hypotenuse of the Pythagorean triple 336-448-560. What is the greatest common factor of those three numbers?

560 Puzzle

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

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  • 560 is a composite number.
  • Prime factorization: 560 = 2 x 2 x 2 x 2 x 5 x 7, which can be written 560 = (2^4) x 5 x 7
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 560 has exactly 20 factors.
  • Factors of 560: 1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 28, 35, 40, 56, 70, 80, 112, 140, 280, 560
  • Factor pairs: 560 = 1 x 560, 2 x 280, 4 x 140, 5 x 112, 7 x 80, 8 x 70, 10 x 56, 14 x 40, 16 x 35 or 20 x 28
  • Taking the factor pair with the largest square number factor, we get √560 = (√16)(√35) = 4√35 ≈ 23.664319

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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.

560 Factors