Pythagorean Triple

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

584 and Level 6

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584 is the hypotenuse of the Pythagorean triple 384-440-584. Which factor of 584 is the greatest common factor of those three numbers?

584 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-10

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  • 584 is a composite number.
  • Prime factorization: 584 = 2 x 2 x 2 x 73, which can be written 584 = (2^3) x 73
  • 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 584 has exactly 8 factors.
  • Factors of 584: 1, 2, 4, 8, 73, 146, 292, 584
  • Factor pairs: 584 = 1 x 584, 2 x 292, 4 x 146, or 8 x 73
  • Taking the factor pair with the largest square number factor, we get √584 = (√4)(√146) = 2√146 ≈ 24.16609

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584 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

582 and Level 4

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582 is the sum of all the prime numbers from 59 to 89. Can you list those eight consecutive primes?

582 is also the hypotenuse of the Pythagorean triple 390-432-582. Which of 582’s factors is the greatest common factor of those three numbers?

582 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-10

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  • 582 is a composite number.
  • Prime factorization: 582 = 2 x 3 x 97
  • 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 582 has exactly 8 factors.
  • Factors of 582: 1, 2, 3, 6, 97, 194, 291, 582
  • Factor pairs: 582 = 1 x 582, 2 x 291, 3 x 194, or 6 x 97
  • 582 has no square factors that allow its square root to be simplified. √582 ≈ 24.124676

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582 Logic

580 and Level 2

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580 is the sum of all the prime numbers from 83 to 107. Do you know what those consecutive prime numbers are?

580 is also the hypotenuse of four Pythagorean triples. In each case can you find the factor of 580 that is the greatest common factor of the triple?

  • 68-576-580
  • 96-572-580
  • 348-464-580
  • 400-420-580

580 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-10

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  • 580 is a composite number.
  • Prime factorization: 580 = 2 x 2 x 5 x 29, which can be written 580 = (2^2) x 5 x 29
  • 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 580 has exactly 12 factors.
  • Factors of 580: 1, 2, 4, 5, 10, 20, 29, 58, 116, 145, 290, 580
  • Factor pairs: 580 = 1 x 580, 2 x 290, 4 x 145, 5 x 116, 10 x 58, or 20 x 29
  • Taking the factor pair with the largest square number factor, we get √580 = (√4)(√145) = 2√145 ≈ 24.083189

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580 Factors

579 and Level 1

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579 is the hypotenuse of the Pythagorean triple 285-504-579. Which of 579’s factors is the greatest common factor of those three numbers?

Last week someone googled “find,the least 6 digit which has 173 factor” and arrived at Findthefactors.com. Here how to find the answer: Divide 100,000 by 173 and get 578.03 approximately. Round that answer up to 579. Multiply 579 by 173 and get 100167, the smallest 6-digit number that has 173 as a factor.

579 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-08-10

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  • 579 is a composite number.
  • Prime factorization: 579 = 3 x 193
  • 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 579 has exactly 4 factors.
  • Factors of 579: 1, 3, 193, 579
  • Factor pairs: 579 = 1 x 579 or 3 x 193
  • 579 has no square factors that allow its square root to be simplified. √579 ≈ 24.0624188

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579 Factors

578 When I stopped teaching about thousands, it made a BIG difference

/ ivasallay / 6 Comments

Several weeks ago I helped a young girl with a place value page in her workbook. The largest number she needed to write out in words was less than 900 million.

The young girl was very confused so I explained it exactly the same way the book explained it – which was exactly the same way it was explained to me when I was a kid.

She easily understood the ones place, the tens place, the hundreds place, and the thousands place.  The trouble began when we started working with the ten thousands place and became even worse when I started talking about the hundred thousands place, the millions place, the ten millions place, and the hundred millions place.

The young girl could easily and correctly separate the digits of any multi-digit number into groups of three, but reading or writing that number using words baffled her.

Also translating number words back into digits and putting those digits into the right places was equally challenging for her.

After explaining what to do on EVERY single problem, I knew she really didn’t get it….even though the assignment was finally finished.

Then last week on twitter I saw a retweet of this:

I wondered what crazy, radical thing was meant by that statement so I clicked on the link and read a very clear and easy-to-understand explanation of how to teach place value. I was very impressed.

This last Friday I saw that young girl again, and I said, “I want to show you something.”

I took my red pen and wrote in her workbook,

place value places

and this time I actually taught her the concept of place value. This time she got it!

Triumphantly she wrote down a 14-digit number made from some “random” numbers that popped into her head and then read it to me perfectly. It was as thrilling for me as it was for her!

An hour before she couldn’t correctly read 90% of the whole numbers less than a million, but now she had MASTERED thousands, millions, billions, and even trillions.

I really like that blogger/tweeter Michael Tidd chose “units” as the last category because that is the natural place to say miles, meters, dollars or whatever the unit happens to be. The unit this young girl chose for her 14-digit number was CATS.

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578 is the hypotenuse of two Pythagorean triples: 322-480-578 and 272-510-578. Each triple has a greatest common factor. Which factors of 578 could they be?

  • 578 is a composite number.
  • Prime factorization: 578 = 2 x 17 x 17, which can be written 578 = 2 x (17^2)
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 x 3  = 6. Therefore 578 has exactly 6 factors.
  • Factors of 578: 1, 2, 17, 34, 289, 578
  • Factor pairs: 578 = 1 x 578, 2 x 289, or 17 x 34
  • Taking the factor pair with the largest square number factor, we get √578 = (√289)(√2) = 17√2 ≈ 24.04163

578 cake

2 x 17 x 17 = 578

575 and Level 4

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575 is the hypotenuse of two Pythagorean triples: 345-460-575 and 161-552-575. What is the greatest common factor of each of those triples?

575 = 23 x 25, which can be written (24 – 1)(24 + 1) or (24^2) – 1.

575 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-03

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  • 575 is a composite number.
  • Prime factorization: 575 = 5 x 5 x 23, which can be written 575 = (5^2) x 23
  • 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 575 has exactly 6 factors.
  • Factors of 575: 1, 5, 23, 25, 115, 575
  • Factor pairs: 575 = 1 x 575, 5 x 115, or 23 x 25
  • Taking the factor pair with the largest square number factor, we get √575 = (√25)(√23) = 3√23 ≈ 23.9791576

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575 Logic

572 and Level 1

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72 can be evenly divided by 4 so 572 is divisible by 4.

5 – 7 + 2 = 0, which can be evenly divided by 11, so 11 is a factor of 572.

572 is the hypotenuse of the Pythagorean triple 220-528-572. Which factor of 572 is the greatest common factor of those three numbers?

572 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-08-03

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  • 572 is a composite number.
  • Prime factorization: 572 = 2 x 2 x 11 x 13, which can be written 572 = (2^2) x 11 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 572 has exactly 12 factors.
  • Factors of 572: 1, 2, 4, 11, 13, 22, 26, 44, 52, 143, 286, 572
  • Factor pairs: 572 = 1 x 572, 2 x 286, 4 x 143, 11 x 52, 13 x 44, or 22 x 26
  • Taking the factor pair with the largest square number factor, we get √572 = (√4)(√143) = 2√143 ≈ 23.91652

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572 Factors

570 and Level 6

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570 is the hypotenuse of Pythagorean triple 342-456-570. What 3 digit number is the greatest common factor of those three numbers?

570 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-07-27

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  • 570 is a composite number.
  • Prime factorization: 570 = 2 x 3 x 5 x 19
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 570 has exactly 16 factors.
  • Factors of 570: 1, 2, 3, 5, 6, 10, 15, 19, 30, 38, 57, 95, 114, 190, 285, 570
  • Factor pairs: 570 = 1 x 570, 2 x 285, 3 x 190, 5 x 114, 6 x 95, 10 x 57, 15 x 38, or 19 x 30
  • 570 has no square factors that allow its square root to be simplified. √570 ≈ 23.87467

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570 Logic