Month: August 2015

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

577 and Level 6

/ ivasallay / Leave a comment

24² + 1 = 577 means 577 is the hypotenuse of a primitive Pythagorean triple which happens to be 48-575-577.

577 Puzzle

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

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  • 577 is a prime number.
  • Prime factorization: 577 is prime.
  • The exponent of prime number 577 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 577 has exactly 2 factors.
  • Factors of 577: 1, 577
  • Factor pairs: 577 = 1 x 577
  • 577 has no square factors that allow its square root to be simplified. √577 ≈ 24.020824

How do we know that 577 is a prime number? If 577 were not a prime number, then it would be divisible by at least one prime number less than or equal to √577 ≈ 24.02. Since 577 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 577 is a prime number.

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577 Logic

576 and Level 5

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The number formed from the last 2 digits of 576 is divisible by 4. That means 576 is also divisible by 4.

576 is made from 3 consecutive numbers so it is divisible by 3. Since the middle number, 6, is also divisible by 3, we know that 576 is also divisible by 9.

Either of the previous statements is enough to indicate that √576 can be reduced. In fact, boy, can it ever be reduced!

576 is the smallest number to have exactly 21 factors. (Only perfect squares have an odd number of factors.)

576 is also the sum of consecutive primes in two different ways:

  • 283 + 293 = 576
  • 137 + 139 + 149 + 151 = 576

576 Puzzle

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

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  • 576 is a composite number.
  • Prime factorization: 576 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3, which can be written 576 = (2^6) x (3^2)
  • The exponents in the prime factorization are 6 and 2. Adding one to each and multiplying we get (6 + 1)(2 + 1) = 7 x 3 = 21. Therefore 576 has exactly 21 factors.
  • Factors of 576: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 576
  • Factor pairs: 576 = 1 x 576, 2 x 288, 3 x 192, 4 x 144, 6 x 96, 8 x 72, 9 x 64, 12 x 48, 16 x 36, 18 x 32 or 24 x 24
  • 576 is a perfect square. √576 = 24

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576 Logic

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

574 Start at the top of the first column and fly down one cell at a time to solve this Level 3 puzzle

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One of 574’s factors is the hypotenuse of a primitive Pythagorean triple so 574 is the hypotenuse of the triple 126-560-574. Which of 574’s factors is the greatest common factor of those three numbers?

574 Puzzle

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

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  • 574 is a composite number.
  • Prime factorization: 574 = 2 x 7 x 41
  • 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 574 has exactly 8 factors.
  • Factors of 574: 1, 2, 7, 14, 41, 82, 287, 574
  • Factor pairs: 574 = 1 x 574, 2 x 287, 7 x 82, or 14 x 41
  • 574 has no square factors that allow its square root to be simplified. √574 ≈ 23.958297.

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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.

574 Factors

573 and Level 2

/ ivasallay / Leave a comment

573 is made from three consecutive odd numbers so it is divisible by 3. The number in the middle, 5, is not divisible by 3, so 573 is not divisible by 9.

573 squared is 328,329, a 6 digit number that looks like two consecutive numbers! Thank you OEIS.org for that interesting fact.

573 Puzzle

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

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  • 573 is a composite number.
  • Prime factorization: 573 = 3 x 191
  • 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 573 has exactly 4 factors.
  • Factors of 573: 1, 3, 191, 573
  • Factor pairs: 573 = 1 x 573 or 3 x 191
  • 573 has no square factors that allow its square root to be simplified. √573 ≈ 23.9374

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573 Factors

572 and Level 1

/ ivasallay / Leave a comment

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

571 Family Time

/ ivasallay / 5 Comments

For the last two weeks we’ve been spending time with our grandchildren in Salt Lake City, Utah; Portland, Oregon; Houston, Texas; and Hartford, Connecticut. We also spent time with my brother’s daughter and her two children. AND we celebrated my sister’s 70th birthday party and reconnected with all of her children in Portland. We have been busy, and I am tired, but happy.

We’ve had a wonderful time with all of them, and it was so painful to say goodbye. We and a few other people took pictures except we didn’t get any pictures of the grandkids that live within 10 miles of our home. What’s with that? I’m as guilty as anyone else who takes people and things for granted.

Here’s a picture of me holding my newest grandson. My daughter is in the background holding one of her nieces.

 

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Since 171 + 190 + 210 = 571, it is the sum of the 18th, 19th, and 20th triangular numbers. That makes 571 the 20th centered triangular number.

  • 571 is a prime number and a twin prime with 569.
  • Prime factorization: 571 is prime.
  • The exponent of prime number 571 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 571 has exactly 2 factors.
  • Factors of 571: 1, 571
  • Factor pairs: 571 = 1 x 571
  • 571 has no square factors that allow its square root to be simplified. √571 ≈ 23.8956

How do we know that 571 is a prime number? If 571 were not a prime number, then it would be divisible by at least one prime number less than or equal to √571 ≈ 23.9. Since 571 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 571 is a prime number.