383 is in only one Pythagorean triple, and it’s primitive

/ ivasallay / Leave a comment

A Primitive Threesome https://www.mycomicshop.com/search?TID=243401

Although it would take awhile to count to 383, it is still a relatively small number. If you scroll down, you will see that it is also a prime number. Since all odd numbers greater than one are the short legs of a primitive Pythagorean triple, 383 is also. I will explain how to find that primitive triple using the Flintstones, the primitive yet modern stone age family as an analogy. You can use this method on ANY odd number greater than one to find a primitive triple:

  • 383 is like Pebbles the shortest one in the family
  • If we round down (383^2)/2 or 73344.5 to 73,344, we get a much bigger number, much like Fred is much taller than Pebbles in the family
  • If we round the same expression up, we get 73,345, much like Wilma is just a little taller than Fred.
  • 383 – 73344 – 73345 is a primitive Pythagorean triple just as Pebbles, Fred, and Wilma are a primitive yet modern stone age family of three. Test it out: (383^2) + (73344^2) = (73345^2)

Prime numbers can be one of the legs of only one Pythagorean triple, and it will always be a primitive. Sometimes prime numbers are the hypotenuse of a primitive Pythagorean triple, but 383 isn’t. In a future post, I’ll explain how to determine if a prime number is ever the hypotenuse.

Here is 383’s factoring information:

  • 383 is a prime number.
  • Prime factorization: 383 is prime.
  • The exponent of prime number 383 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 383 has exactly 2 factors.
  • Factors of 383: 1, 383
  • Factor pairs: 383 = 1 x 383
  • 383 has no square factors that allow its square root to be simplified. √383 ≈ 19.570

How do we know that 383 is a prime number? If 383 were not a prime number, then it would be divisible by at least one prime number less than or equal to √383 ≈ 19.570. Since 383 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 383 is a prime number.

382 and Level 5

/ ivasallay / Leave a comment

382 is another even composite number. Scroll past the puzzle for its factoring information.

382 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-02

  • 382 is a composite number.
  • Prime factorization: 382 = 2 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 382 has exactly 4 factors.
  • Factors of 382: 1, 2, 191, 382
  • Factor pairs: 382 = 1 x 382 or 2 x 191
  • 382 has no square factors that allow its square root to be simplified. √382 ≈ 19.545

382 is in only one Pythagorean triple: 382 – 36480 – 36482 which is 2 times (191 – 18240 – 18241).

382 Logic

381 and Level 4

/ ivasallay / Leave a comment

Is 381 Prime or composite? Its digits can be grouped into 3 and 81. Both groups are divisible by 3 so 381 is composite. Scroll down past the puzzle to view its factors.

381 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-02

  • 381 is a composite number.
  • Prime factorization: 381 = 3 x 127
  • 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 381 has exactly 4 factors.
  • Factors of 381: 1, 3, 127, 381
  • Factor pairs: 381 = 1 x 381 or 3 x 127
  • 381 has no square factors that allow its square root to be simplified. √381 ≈ 19.519

Pythagorean triples:

  • 381 – 508 – 635 which is 127 times (3 – 4 – 5 )
  • 381 – 24192 – 24195 which is 3 times (127 – 8064 – 8065)
  • 381 – 8060 – 8069, a primitive
  • 381 – 72580 – 72581, a primitive

381 Logic

380 and Level 3

/ ivasallay / Leave a comment

380 is even so it has to be a composite number. Scroll down past the puzzle for all of its factors.

380 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-02

  • 380 is a composite number.
  • Prime factorization: 380 = 2 x 2 x 5 x 19, which can be written 380 = (2^2) x 5 x 19
  • 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 380 has exactly 12 factors.
  • Factors of 380: 1, 2, 4, 5, 10, 19, 20, 38, 76, 95, 190, 380
  • Factor pairs: 380 = 1 x 380, 2 x 190, 4 x 95, 5 x 76, 10 x 38, or 19 x 20
  • Taking the factor pair with the largest square number factor, we get √380 = (√4)(√95) = 2√95 ≈ 19.494

Since those consecutive numbers in the last factor pair are the last possibility on the chart, we know that 380 is the sum of the first 19 even numbers. Thus,

2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 = 380. Those numbers can be rearranged this way:

2 + 36 + 4 + 34 + 6 + 32 + 8 + 30 + 10 + 28 + 12 + 26 + 14 + 24 + 16 + 22 + 18 + 20 + 38 = 380, which is more easily seen as the sum of ten 38’s.

A Logical Approach to FIND THE FACTORS: 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.

 380 Factors

379 and Level 2

/ ivasallay / Leave a comment

2^379 (mod 379) = 2 so 379 is VERY LIKELY a prime number. Scroll down below the puzzle to know for sure.

Alan Parr, who worked on some fun projects at NRICH, recently wrote about his game, Envelopes, on his blog. This easy-to-make game can be used to reinforce addition or multiplication facts, and it never looks anything like a worksheet.

379 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-02

  • 379 is a prime number.
  • Prime factorization: 379 is prime.
  • The exponent of prime number 379 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 379 has exactly 2 factors.
  • Factors of 379: 1, 379
  • Factor pairs: 379 = 1 x 379
  • 379 has no square factors that allow its square root to be simplified. √379 ≈ 19.468

How do we know that 379 is a prime number? If 379 were not a prime number, then it would be divisible by at least one prime number less than or equal to √379 ≈ 19.468. Since 379 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 379 is a prime number.

379 Factors

378 Do You See a Pattern?

/ ivasallay / 5 Comments

Most people know that 378 is a composite number because it’s obviously divisible by 2, but I know that 378 is a composite number because 18 x 21 = 378. I didn’t use a calculator or multiply 18 and 21 by hand. I know this multiplication fact because I can see a pattern, because 18 + 3 = 21 and because SUBTRACTING 2 from a number is easy.

“What is this pattern?” you ask. Here it is. Do you see the pattern, too?

2nd Multiplication Pattern

The pattern looks like this on the multiplication table:

Basic Multiplication Table Pattern

So how did I know that 18 x 21 = 378? I knew because I can easily compute 19 x 20 = 380 in my head, and 380 – 2 = 378.

What is all the factoring information for 378?

  • 378 is a composite number.
  • Prime factorization: 378 = 2 x 3 x 3 x 3 x 7, which can be written 378 = 2 x (3^3) x 7
  • 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 378 has exactly 16 factors.
  • Factors of 378: 1, 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, 189, 378
  • Factor pairs: 378 = 1 x 378, 2 x 189, 3 x 126, 6 x 63, 7 x 54, 9 x 42, 14 x 27, or 18 x 21
  • Taking the factor pair with the largest square number factor, we get √378 = (√9)(√42) = 3√42 ≈ 18.442

 

377 and Level 1

/ ivasallay / 1 Comment

377 isn’t even, and the sum of its digits don’t add up to a multiple of 3. I’ll give it the quick prime number test: 2^377 (mod 377) = 188, not 2, so 377 is definitely a composite number. Scroll down past the puzzle to see its factoring information.

 377 Puzzle

 Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-02

  • 377 is a composite number.
  • Prime factorization: 377 = 13 x 29
  • 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 377 has exactly 4 factors.
  • Factors of 377: 1, 13, 29, 377
  • Factor pairs: 377 = 1 x 377 or 13 x 29
  • 377 has no square factors that allow its square root to be simplified. √377 ≈ 19.416

13 and 29 are both Pythagorean triple hypotenuses so 377 can be written as the sum of two squares two different ways, and it is the hypotenuse of FOUR Pythagorean triples, half of which are primitives:

  • 19² + 4² = 377
  • 16² + 11² = 377
  • 135² + 352² = 377²
  • 145² + 348² = 377²
  • 152² + 345² = 377²
  • 260² + 273² = 377²

376 Happy Birthday, Doug

/ ivasallay / Leave a comment

I hope you have a wonderful day, Doug.

In our family we were the closest in age, so we often went to the same school. He was brilliant in mathematics, and I wanted to follow in his footsteps. Before I started this blog, I showed him and his wife a few Find the Factors puzzles. They were both very encouraging. We live thousands of miles away from each other, but I am grateful we were able to spend time with each other twice in 2014.

Happy birthday, Doug!

376 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-02

  • 376 is a composite number.
  • Prime factorization: 376 = 2 x 2 x 2 x 47, which can be written 376 = (2^3) x 47
  • 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 376 has exactly 8 factors.
  • Factors of 376: 1, 2, 4, 8, 47, 94, 188, 376
  • Factor pairs: 376 = 1 x 376, 2 x 188, 4 x 94, or 8 x 47
  • Taking the factor pair with the largest square number factor, we get √376 = (√4)(√94) = 2√94 ≈ 19.391

376 Logic

Here’s an interesting tweet about the number 376:

//platform.twitter.com/widgets.js

375 and Level 6

/ ivasallay / Leave a comment

375 is a composite number because its last digit is 5. Scroll down below the puzzle to see its factoring information.

375 Puzzle

Print the puzzles or type the factors on this excel file:12 Factors 2015-01-26

  • 375 is a composite number.
  • Prime factorization: 375 = 3 x 5 x 5 x 5, which can be written 375 = 3 x (5^3)
  • The exponents in the prime factorization are 1 and 3. Adding one to each and multiplying we get (1 + 1)(3 + 1) = 2 x 4 = 8. Therefore 375 has exactly 8 factors.
  • Factors of 375: 1, 3, 5, 15, 25, 75, 125, 375
  • Factor pairs: 375 = 1 x 375, 3 x 125, 5 x 75, or 15 x 25
  • Taking the factor pair with the largest square number factor, we get √375 = (√15)(√25) = 5√15 ≈ 19.365

375 is in this cool pattern:

375 Logic

374 and Level 5

/ ivasallay / Leave a comment

374 is even so it is a composite number. Its factoring information is below the puzzle.

374 Puzzle

Print the puzzles or type the factors on this excel file:12 Factors 2015-01-26

  • 374 is a composite number.
  • Prime factorization: 374 = 2 x 11 x 17
  • 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 374 has exactly 8 factors.
  • Factors of 374: 1, 2, 11, 17, 22, 34, 187, 374
  • Factor pairs: 374 = 1 x 374, 2 x 187, 11 x 34, or 17 x 22
  • 374 has no square factors that allow its square root to be simplified. √374 ≈ 19.339

374 Logic