Composite number

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

 

542 and Level 6

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Here are some ways to make 542 by adding together exactly three square numbers.

  • 542 = (21^2) + (10^2) + (1^2)
  • 542 = (19^2) + (10^2) + (9^2)
  • 542 = (18^2) + (13^2) + (7^2)
  • 542 = (15^2) + (14^2) + (11^2)

542 Puzzle

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

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  • 542 is a composite number.
  • Prime factorization: 542 = 2 x 271
  • 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 542 has exactly 4 factors.
  • Factors of 542: 1, 2, 271, 542
  • Factor pairs: 542 = 1 x 542 or 2 x 271
  • 542 has no square factors that allow its square root to be simplified. √542 ≈ 23.280893

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542 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

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

536 Family Reunion

/ ivasallay / 2 Comments

Last week I attended a family reunion. My uncle Bob showed me a very clever way that helps him remember the number of children that my dad and each of his siblings had.

How Many Children

In case you are wondering, I was one of Leonard’s fifteen kids: He and his first wife had 4 children. They divorced. He met my mom who already had a child of her own. They married and had 6 children. She died. Then after he married my step-mother who already had two grown children, they had two more.

  • 536 is a composite number.
  • Prime factorization: 536 = 2 x 2 x 2 x 67, which can be written 536 = (2^3) x 67
  • 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 536 has exactly 8 factors.
  • Factors of 536: 1, 2, 4, 8, 67, 134, 268, 536
  • Factor pairs: 536 = 1 x 536, 2 x 268, 4 x 134, or 8 x 67
  • Taking the factor pair with the largest square number factor, we get √536 = (√4)(√134) = 2√134 ≈ 23.15167

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