594 and Level 2

/ ivasallay / Leave a comment

It is as easy as 1-2-3 to remember this number fact from OEIS.org:

(1^5) + (2^9) + (3^4) = 594

594 is the sum of the ten prime numbers from 41 to 79.

594 is even so it is divisible by 2.

5 + 9 + 4 = 18 so 594 is divisible by both 3 and 9 (and by 6 and 18 because 594 is even)

5 – 9 + 4 = 0 so 594 is divisible by 11.

594 Puzzle

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

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  • 594 is a composite number.
  • Prime factorization: 594 = 2 x 3 x 3 x 3 x 11, which can be written 594 = 2 x (3^3) x 11
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 x 4 x 2 = 16. Therefore 594 has exactly 16 factors.
  • Factors of 594: 1, 2, 3, 6, 9, 11, 18, 22, 27, 33, 54, 66, 99, 198, 297, 594
  • Factor pairs: 594 = 1 x 594, 2 x 297, 3 x 198, 6 x 99, 9 x 66, 11 x 54, 18 x 33, or 22 x 27
  • Taking the factor pair with the largest square number factor, we get √594 = (√9)(√66) = 3√66 ≈ 24.372115

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594 Factors

593 and Level 1

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593 is the sum of the seven prime numbers from 71 to 101 as well as the nine prime numbers from 47 to 83.

When the number formed from the last two digits of prime number 593 is divided by 4, the remainder is 1 so 593 can be expressed at the sum of two squares AND is the hypotenuse of a primitive Pythagorean triple. Specifically:

23² + 8² = 593

593 is the hypotenuse of the primitive Pythagorean triple prime 368-465-593

593 Puzzle

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

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  • 593 is a prime number.
  • Prime factorization: 593 is prime.
  • The exponent of prime number 593 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 593 has exactly 2 factors.
  • Factors of 593: 1, 593
  • Factor pairs: 593 = 1 x 593
  • 593 has no square factors that allow its square root to be simplified. √593 ≈ 24.35159

How do we know that 593 is a prime number? If 593 were not a prime number, then it would be divisible by at least one prime number less than or equal to √593 ≈ 24.4. Since 593 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 593 is a prime number.

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593 Factors

592 and The Broken Internet Connection

/ ivasallay / 4 Comments

There is a lot of road construction near my house. I don’t know if it is related, but the outside cable that provides us internet connection died sometime on the morning of Sunday, 23 August 2015.

That broken cable line meant that we had no home phone service, no internet, no banking information, no twitter, no facebook = no looking at the latest pictures of my grandchildren, no reading Blogspot, no watching Netflix or Youtube, and no reading or writing on WordPress.

I don’t have a smart phone, but my husband uses the Wifi in our house for his. His smart phone wasn’t smart enough to switch to our phone’s data plan. Instead it relentlessly tried to connect to the Wifi every time either of us tried to use it.

My dead internet connection meant that if you were looking for information about the number 592 (and who isn’t), you had to wait a few days:

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592 is a palindrome in three bases:

  • 727 BASE 9 because 7(81) + 2(9) + 7(1) = 592
  • 414 BASE 12 because 4(144) + 1(12) + 4(1) = 592
  • GG BASE 36 (G is 16 in base 10) because 16(36) + 16(1) = 592

592 is the hypotenuse of the Pythagorean triple 192-560-592. Which factor of 592 is the greatest common factor of those three numbers?

  • 592 is a composite number.
  • Prime factorization: 592 = 2 x 2 x 2 x 2 x 37, which can be written 592 = (2^4) x 37
  • 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 592 has exactly 10 factors.
  • Factors of 592: 1, 2, 4, 8, 16, 37, 74, 148, 296, 592
  • Factor pairs: 592 = 1 x 592, 2 x 296, 4 x 148, 8 x 74, or 16 x 37
  • Taking the factor pair with the largest square number factor, we get √592 = (√16)(√37) = 4√37 ≈ 24.33105

591 and a Sometimes Impossible Level 6

/ ivasallay / 2 Comments

5 + 9 + 1 = 15, a multiple of 3 so 591 is divisible by 3.

However, since 9 is a multiple of 3, it isn’t necessary to include it in the equation, so I simple say 5 + 1 = 6, a multiple of 3 so 591 is divisible by 3.

591 is the hypotenuse of the Pythagorean triple 84-585-591. Which of 591’s factors is the greatest common factor of those three numbers?

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I love that Joseph Nebus wrote that the FIND THE FACTORS puzzles are “a fun recreational mathematics puzzle even if the level 6’s and sometimes level 5’s will sometimes feel impossible.”

Judge for yourself. Is this Level 6 puzzle impossible or not!

591 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-17

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  • 591 is a composite number.
  • Prime factorization: 591 = 3 x 197
  • 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 591 has exactly 4 factors.
  • Factors of 591: 1, 3, 197, 591
  • Factor pairs: 591 = 1 x 591 or 3 x 197
  • 591 has no square factors that allow its square root to be simplified. √591 ≈ 24.31049

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591 Logic

 

590 and Level 5

/ ivasallay / Leave a comment

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

589 and Level 4

/ ivasallay / 2 Comments

589 is the sum of the three prime numbers from 193 to 199. It should be pretty obvious what that middle prime number is.

589 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-17

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  • 589 is a composite number.
  • Prime factorization: 589 = 19 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 589 has exactly 4 factors.
  • Factors of 589: 1, 19, 31, 589
  • Factor pairs: 589 = 1 x 589 or 19 x 31
  • 589 has no square factors that allow its square root to be simplified. √589 ≈ 24.269322

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589 Logic

Simplifying √588 and Level 3

/ ivasallay / Leave a comment

88 is divisible by 4 so 588 is also divisible by 4, and that means √588 can be reduced.

About 83% of the numbers that have reducible square roots are divisible by 4 and/or by 9, and it is so easy to tell if even a very long number is divisible by either of those numbers. It is also easier to divide a number by 4 or 9 than it is to divide by their square roots twice.

When I reduce a square root, I like to make a little cake and start by dividing by 100, 4, or 9 if any of those numbers are its factors. Here are the steps I used to make a cake for 588 with as many perfect squares on the outside of the cake as possible.

  1. 588 ÷ 4 = 147
  2. 147 is not divisible by 4 again, but 5 + 8 + 8 = 21 so 147 is divisible by 3, but not by 9.
  3. 147 ÷ 3 = 49 which is a perfect square, so I stop dividing and simply take the square roots of everything on the outside of the cake and multiply them together.

This is what my cake looks like:

588 cake

And now for today’s Level 3 puzzle:

588 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-17

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  • 588 is a composite number.
  • Prime factorization: 588 = 2 x 2 x 3 x 7 x 7, which can be written 588 = (2^2) x 3 x (7^2)
  • The exponents in the prime factorization are 2, 1 and 2. Adding one to each and multiplying we get (2 + 1)(1 + 1)(2 + 1) = 3 x 2 x 3 = 18. Therefore 588 has exactly 18 factors.
  • Factors of 588: 1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 49, 84, 98, 147, 196, 294, 588
  • Factor pairs: 588 = 1 x 588, 2 x 294, 3 x 196, 4 x 147, 6 x 98, 7 x 84, 12 x 49, 14 x 42 or 21 x 28
  • Taking the factor pair with the largest square number factor, we get √588 = (√196)(√3) = 14√3 ≈ 24.248711

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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.

588 Factors

587 and Level 2

/ ivasallay / Leave a comment

587 is the sum of the five prime numbers from 107 to 131.

587 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-17

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  • 587 is a prime number.
  • Prime factorization: 587 is prime.
  • The exponent of prime number 587 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 587 has exactly 2 factors.
  • Factors of 587: 1, 587
  • Factor pairs: 587 = 1 x 587
  • 587 has no square factors that allow its square root to be simplified. √587 ≈ 24.22808

How do we know that 587 is a prime number? If 587 were not a prime number, then it would be divisible by at least one prime number less than or equal to √587 ≈ 24.2. Since 587 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 587 is a prime number.

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587 Factors

586 and Level 1

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586 is the hypotenuse of the Pythagorean triple 136-570-586. Which factor of 586 is the greatest common factor of those three numbers?

586 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-17

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  • 586 is a composite number.
  • Prime factorization: 586 = 2 x 293
  • 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 586 has exactly 4 factors.
  • Factors of 586: 1, 2, 293, 586
  • Factor pairs: 586 = 1 x 586 or 2 x 293
  • 586 has no square factors that allow its square root to be simplified. √586 ≈ 24.2074

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586 Factors

What can you see in the number 585?

/ ivasallay / 1 Comment

This week I watched an excellent video titled 5 x 9 is more than 45. Indeed 45 is so much more than simply 5 x 9. Every multiplication fact is much more than that mere fact, but Steve Wyborney used 5 x 9 = 45 in his video… Guess what! 585 is a multiple of 45.

As I thought about the number 585, I marveled at some of the hidden mysteries this number holds.

Since 585 is divisible by two different centered square numbers, 5 and 13, I saw that 585 could be represented by this lovely array that has 45 larger squares made up of 13 smaller colorful squares. When you look at the array, do you just see 585 squares or can you see even more multiplication and division facts? If you rotate the array 90 degrees, do the facts change?

585 Squares-1

What do you see in this array of 117 medium sized squares made up of 5 smaller squares:

585 Squares-2

Or this more simple looking array of sixty-five 3 x 3 squares.

585 Squares-3

All of these arrays are in just two dimensions. A 5 x 9 x 13 rectangular prism is ONE way to represent 585 in three dimensions.

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Since factors 5, 13, and 65 are hypotenuses of primitive Pythagorean triples, 585 is the hypotenuse of four Pythagorean triples. Each triple has a different greatest common factor. Can you figure out what each one is?

  • 144-567-585
  • 225-540-585
  • 297-504-585
  • 351-468-585

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And if all of that wasn’t enough, OEIS.org informs us that 585 is a palindrome in 3 different bases!

  • 585 = 1001001001 in base 2
  • 585 = 1111 in base 8
  • 585 = 585 in base 10

Here are two different methods of determining 585 in base 8. In both methods the base 8 representation is in blue.

585 base 8

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  • 585 is a composite number.
  • Prime factorization: 585 = 3 x 3 x 5 x 13, which can be written 585 = (3^2) x 5 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 585 has exactly 12 factors.
  • Factors of 585: 1, 3, 5, 9, 13, 15, 39, 45, 65, 117, 195, 585
  • Factor pairs: 585 = 1 x 585, 3 x 195, 5 x 117, 9 x 65, 13 x 45, or 15 x 39
  • Taking the factor pair with the largest square number factor, we get √585 = (√9)(√65) = 3√65 ≈ 24.18677