logic

540 and Level 4

/ ivasallay / 2 Comments

Today’s Puzzle:

540 Puzzle

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

Here is a logical order to use the clues to solve the puzzle:

540 Logic

Factors of 540:

  • 540 is a composite number.
  • Prime factorization: 540 = 2 x 2 x 3 x 3 x 3 x 5, which can be written 540 = (2^2) x (3^3) x 5
  • The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 x 4 x 2 = 24. Therefore 540 has exactly 24 factors.
  • Factors of 540: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 27, 30, 36, 45, 54, 60, 90, 108, 135, 180, 270, 540
  • Factor pairs: 540 = 1 x 540, 2 x 270, 3 x 180, 4 x 135, 5 x 108, 6 x 90, 9 x 60, 10 x 54, 12 x 45, 15 x 36, 18 x 30 or 20 x 27
  • Taking the factor pair with the largest square number factor, we get √540 = (√36)(√15) = 6√15 ≈ 23.237900077

Sum-Difference Puzzles:

60 has six factor pairs. One of those pairs adds up to 17, and  another one subtracts to 17. Put the factors in the appropriate boxes in the first puzzle.

540 has twelve factor pairs. One of the factor pairs adds up to 51, and a different one subtracts to 51. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

More about the Number 540:

540 is the sum of the fourteen consecutive prime numbers from 13 to 67. Can you list all those prime numbers? It is also the sum of consecutive prime numbers 269 and 271.

540 has the same number of factors as 504. Both of those numbers tie with 360, 420, and 480 for the most factors so far.

540 is the hypotenuse of the Pythagorean triple 324-432-540. What is the greatest common factor of those three numbers?

540 is also an untouchable number.

The sum of the interior angles of every convex pentagon total 540 degrees.

Simplifying √539 and Level 3

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If you add up all the prime numbers from 29 to 71, the sum will be 539.

Specifically, 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 = 539.

How to reduce √539: Since 539 cannot be evenly divided by 100, 4, 9, or 25, look for the smallest prime number that will divide into 539. You will soon note that 539 ÷ 7 = 77. Divide 77 by 7 again to get 11. I like to make a little cake that looks like this:

539 cake

Then I take the square root of everything on the outside of the cake: √539 = (√7)(√7)(√11) = 7√11

539 Puzzle

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

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  • 539 is a composite number.
  • Prime factorization: 539 = 7 x 7 x 11, which can be written 539 = (7^2) x 11
  • 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 539 has exactly 6 factors.
  • Factors of 539: 1, 7, 11, 49, 77, 539
  • Factor pairs: 539 = 1 x 539, 7 x 77, or 11 x 49
  • Taking the factor pair with the largest square number factor, we get √539 = (√49)(√11) = 7√11 ≈ 23.21637353

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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.

539 Factors

538 and Level 2

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538 is the hypotenuse of the Pythagorean triple 138-520-538. Can you find the greatest common factor of those three numbers?

538 Puzzle

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

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  • 538 is a composite number.
  • Prime factorization: 538 = 2 x 269
  • 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 538 has exactly 4 factors.
  • Factors of 538: 1, 2, 269, 538
  • Factor pairs: 538 = 1 x 538 or 2 x 269
  • 538 has no square factors that allow its square root to be simplified. √538 ≈ 23.194827

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538 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

535 and Level 6

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535 is the hypotenuse of the Pythagorean triple 321-428-535. Can you find the greatest common factor of those three numbers?

535 Puzzle

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

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  • 535 is a composite number.
  • Prime factorization: 535 = 5 x 107
  • 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 535 has exactly 4 factors.
  • Factors of 535: 1, 5, 107, 535
  • Factor pairs: 535 = 1 x 535 or 5 x 107
  • 535 has no square factors that allow its square root to be simplified. √535 ≈ 23.130067

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535 Logic

534 and Level 5

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534 is made of three consecutive digits so it can be evenly divided by three.

534 is the sum of consecutive primes: 127 + 131 + 137 + 139 = 534.

534 is the hypotenuse of the Pythagorean triple 234-480-534. Can you find the greatest common factor of those three numbers?

534 Puzzle

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

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  • 534 is a composite number.
  • Prime factorization: 534 = 2 x 3 x 89
  • 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 534 has exactly 8 factors.
  • Factors of 534: 1, 2, 3, 6, 89, 178, 267, 534
  • Factor pairs: 534 = 1 x 534, 2 x 267, 3 x 178, or 6 x 89
  • 534 has no square factors that allow its square root to be simplified. √534 ≈ 23.108440.

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534 Logic

533 and Level 4

/ ivasallay / 3 Comments

533 is the sum of consecutive primes two different ways: 173 + 179 + 181 = 533 = 101 + 103 + 107 + 109 + 113.

533 = (23^2) + (2^2), and 533 = (22^2) + (7^2)

533 is the hypotenuse of four Pythagorean triples. Some of the triples have a greatest common factor greater than one, and the rest are primitive. Which are which?

  • 92-525-533
  • 117-520-533
  • 205-492-533
  • 308-435-533

533 Puzzle

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

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  • 533 is a composite number.
  • Prime factorization: 533 = 13 x 41
  • 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 533 has exactly 4 factors.
  • Factors of 533: 1, 13, 41, 533
  • Factor pairs: 533 = 1 x 533 or 13 x 41
  • 533 has no square factors that allow its square root to be simplified. √533 ≈ 23.08679

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533 Logic

532 and Level 3

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532 is the sum of consecutive prime numbers 263 and 269.

532 is also the 19th pentagonal number.

532 Puzzle

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

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  • 532 is a composite number.
  • Prime factorization: 532 = 2 x 2 x 7 x 19, which can be written 532 = (2^2) x 7 x 19
  • 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 532 has exactly 12 factors.
  • Factors of 532: 1, 2, 4, 7, 14, 19, 28, 38, 76, 133, 266, 532
  • Factor pairs: 532 = 1 x 532, 2 x 266, 4 x 133, 7 x 76, 14 x 38, or 19 x 28
  • Taking the factor pair with the largest square number factor, we get √532 = (√4)(√133) = 2√133 ≈ 23.065125

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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.

532 Factors

 

531 and Level 2

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5 + 3 + 1 = 9 so 531 is divisible by 3 and by 9. Because it can be evenly divided by 9, it’s square root can be reduced.

531 ÷ 9 = 59, a prime number which obviously has no square factors. Thus √531 = (√9)(√59) = 3√59 in its most reduced form.

531 Puzzle

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

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  • 531 is a composite number.
  • Prime factorization: 531 = 3 x 3 x 59, which can be written 531 = (3^2) x 59
  • 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 531 has exactly 6 factors.
  • Factors of 531: 1, 3, 9, 59, 177, 531
  • Factor pairs: 531 = 1 x 531, 3 x 177, or 9 x 59
  • Taking the factor pair with the largest square number factor, we get √531 = (√9)(√59) = 3√59 ≈ 23.04343724

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531 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