factor pairs

342 and Level 6

/ ivasallay / Leave a comment

  • 342 is a composite number.
  • Prime factorization: 342 = 2 x 3 x 3 x 19, which can be written 342 = 2 x (3^2) x 19
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 342 has exactly 12 factors.
  • Factors of 342: 1, 2, 3, 6, 9, 18, 19, 38, 57, 114, 171, 342
  • Factor pairs: 342 = 1 x 342, 2 x 171, 3 x 114, 6 x 57, 9 x 38, or 18 x 19
  • Taking the factor pair with the largest square number factor, we get √342 = (√9)(√38) = 3√38 ≈ 18.493

342 = 18 × 19, which means it is the sum of the first 18 even numbers.

  • Thus, 2 + 4 + 6 + 8 + . . .  + 32 + 34 + 36 = 342

It also means that we are halfway between 18² and 19², or halfway between 324 and 361. The average of those two numbers is 342.5.

AND it means that 18² + 19² – 1 = 2(342) = 2(18 × 19)

Here are a few things to think about before you try to solve the puzzle I’ve posted below:

  • Will you decide to use the 6 or the 9 as the common factor of 72 and 54?
  • Will 5 or 10 be the common factor of 20 and 50?
  • The common factor for 27 and 9 could be 9 or it could be 3,
  • and the common factor for 32 and 24 is either 4 or 8.
  • Can you use logic to come up with the correct factor choices to make this puzzle work as a multiplication table?

2014-52 Level 6

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-29

2014-52 Level 6 Logic

341 is the smallest composite number that gives a false positive for this Quick Prime Number Test

/ ivasallay / 9 Comments
  • 341 is a composite number.
  • Prime factorization: 341 = 11 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 341 has exactly 4 factors.
  • Factors of 341: 1, 11, 31, 341
  • Factor pairs: 341 = 1 x 341 or 11 x 31
  • 341 has no square factors that allow its square root to be simplified. √341 ≈ 18.466

341 is a composite number that sometimes acts like a prime number. To understand why, we need to understand a little bit about modular arithmetic:

When one number is divided by another, sometimes there is a remainder. Modular arithmetic is all about the remainders. We don’t care how many times one number divides into another number; we only care about the remainder.

Something very curious happens when the equation in the chart below is applied to a prime number greater than 2: The remainder is always 2. It really is! For example, 2^5 = 32 and 32 divided by 5 is 6R2. We say that 32 (mod 5) = 2 or 2^5 (mod 5) = 2 because 2 is the remainder. The fact that the remainder for prime numbers applied to this equation is always 2 is amazing, and can be a QUICK TEST to see if an odd number might be a PRIME NUMBER! If the remainder isn’t 2, that odd number is definitely NOT prime!

Prime Number Test

QUICK PRIME NUMBER TEST (Please, excuse my using = instead of ≡, the “equal sign” that is usually used in modular arithmetic. I think = looks less intimidating.)

Passing the remainder test is a necessary but not a sufficient indicator that a number is prime: Even though 341 is not a prime number, the quantity 2^341 divided by 341 also has a remainder of 2. Since 341 = 11 x 31, but passes the remainder test, it is known as a pseudoprime number.  341 is the smallest composite number that passes this particular test, so 341 is an amazing number!

Also in the chart above, 2 is the most common remainder, followed by 8, then 32, then 128. All of those numbers are odd powers of 2. The even powers of 2 do not appear on the chart at all! That is a very curious phenomenon as well. (However, if x is an even number, it appears that y will usually be an even power of 2.)

Earlier mathematicians have written equivalent expressions and algorithms, but I prefer using “2^x (mod x)” because it takes very few keystrokes to enter into a calculator before hitting the equal sign:

mod 341 calculator

This is only a picture of a calculator.

Look at the image below.  It demonstrates that in prime-numbered rows, the numbers in that row can be divided evenly by that prime number (not counting the 1’s at the beginning and ending of each row).

  • For a composite number, such as 15, at least one of the numbers in the row will NOT be divisible by that composite number. In row 15, 1 is the 0th term, 15 is the 1st term, 105 is the 2nd term, 455 is the 3rd term, and 1365 is the 4th term and so forth.
  • Terms that are divisible by 15 are the 1st term (15), the 2nd term (105), the 4th term (1365), the 7th term (6435), the 8th term (6435), the 11th term (1365), and the 13th term (105). All of those term numbers, 1, 2, 4, 7, 8, 11, and 13, do NOT have factors in common with the number 15.
  • The terms that are NOT divisible by 15 are the 3rd term (455), the 5th term (3003), the 6th term (5005), the 9th term (5005), the 10th term (3003), and the 12th term (455). All of those term numbers, 3, 5, 6, 9, 10, and 12, have at least one factor in common with the number 15.

If we could see the VERY large numbers for the 341st row, and if they weren’t expressed in Scientific Notation, we could note that the following terms would NOT be divisible by 341: terms numbered 11th, 22nd, 33rd, 44th, and so forth and the terms numbered 31st, 62nd, 93rd and so forth. However, if you add those terms together, that very large sum would be divisible by 341. That is so amazing, even though 341 is not prime!

This second image from Pascal’s Triangle demonstrates that the sum of the numbers in any row of Pascal’s triangle equals two raised to the second number in that row.
(There is no second number in the top row, so we could say that second number is zero, and 2º = 1.)

These two images work together so that 2^p (mod p) will always be 2 for every prime number greater than 2 because every number in the prime-numbered rows can be evenly divided by that prime number. (Except the 1 at the beginning of the row and the 1 at the end of the row; Note 1 + 1 = 2)

Now that is why this amazing test for prime numbers works as well as it does while giving just a few false positives!

340 and Level 5

/ ivasallay / Leave a comment

  • 340 is a composite number.
  • Prime factorization: 340 = 2 x 2 x 5 x 17, which can be written 340 = (2^2) x 5 x 17
  • 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 340 has exactly 12 factors.
  • Factors of 340: 1, 2, 4, 5, 10, 17, 20, 34, 68, 85, 170, 340
  • Factor pairs: 340 = 1 x 340, 2 x 170, 4 x 85, 5 x 68, 10 x 34, or 17 x 20
  • Taking the factor pair with the largest square number factor, we get √340 = (√4)(√85) = 2√85 ≈ 18.439

Do you have any goals to improve your thinking power this year? Well, solving puzzles like this one can be good for your brain!

2014-52 Level 5

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-29

2014-52 Level 5 Logic

339 and Level 4

/ ivasallay / Leave a comment

  • 339 is a composite number.
  • Prime factorization: 339 = 3 x 113
  • 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 339 has exactly 4 factors.
  • Factors of 339: 1, 3, 113, 339
  • Factor pairs: 339 = 1 x 339 or 3 x 113
  • 339 has no square factors that allow its square root to be simplified. √339 ≈ 18.412

Existential Spaghetti wrote a blog post a couple of years ago on New Year’s resolutions being yearly-chinese-finger-traps. People tend to choose resolutions that are “both time-consuming and often high-energy.” Ponder that as you solve this Level 4 Chinese Finger Trap Factoring Puzzle made especially for the New Year.

2014-52 Level 4

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-29

 Amazon.com sells these Chinese Finger Traps:

2014-52 Level 4 Logic

338 Liebster Award or Not

/ ivasallay / 12 Comments

  • 338 is a composite number.
  • Prime factorization: 338 = 2 x 13 x 13, which can be written 338 = 2 x 13²
  • 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 338 has exactly 6 factors.
  • Factors of 338: 1, 2, 13, 26, 169, 338
  • Factor pairs: 338 = 1 x 338, 2 x 169, or 13 x 26
  • Taking the factor pair with the largest square number factor, we get √338 = (√2)(√169) = 13√2 ≈ 18.385

Starguy recently nominated me for this blue-green Liebster award. Many bloggers have a no award policy, and I can certainly understand why: receiving an award is a bit of work if you accept it. The Liebster award has been around for a long time, and it has been said that just about every blogger has received it at one time or another.

As these images demonstrate, some recipients have changed the way the award looks.
Since the logo has been changed multiple times by multiple people, I don’t feel bad having a somewhat rebellious attitude as I accept some of the qualifiers for the award while rejecting others. I’m not interested in sharing 11 random facts about myself, answering 11 questions, or coming up with 11 more to ask my nominees. I’m sure you will learn at least 11 things about me just reading this post anyway. (Random fact: I don’t like reading really long posts that go on forever, and this post will be long enough as it is.) If my nominees would like to write and answer questions, they can refer back to the questions asked in Starguy‘s post.

The idea of the Liebster award is to help bloggers get out of their comfort zones and discover new blogs, and I have done that. In fact several of the blogs I mention in this post I discovered within the last couple of weeks.

Rather than make a simple list of nominees, I’m going to do MORE than required and describe one of their noteworthy posts. Hopefully, the way that I do this will cause them to get a pingback notification because rebellious me is probably not going to inform them of their nominations any other way. Besides, if more people click on their links because I nominated them here, then the nomination will be meaningful. They will most likely notice when they view the referrers on their stat pages that Findthefactors is sending several people their way as well. Otherwise, this nomination won’t mean much. Nevertheless, please, click on their links. Here are my 11 nominees in no particular order:

  1. Even though I am an English speaking grandma, I  listened only to Hungarian Christmas carols this year. Justkinga is a Hungarian teenage girl who listens to popular English music all year long. She takes TEN classes at a bilingual school, but somehow finds the time to write a delightful blog in ENGLISH regularly. I ADMIRE HER for writing in English even if there is an occasional grammar error or incorrect word choice. She also writes about Hungarian life and culture.
  2. Established1962 wrote about an ingenious way to make Snakes and Ladders a game of decision making rather than mere chance. In the process, he made the game something even older players would enjoy playing while they observe some subtle mathematics.
  3. Nebushumor wrote a very funny post about an extermination ad that featured an adorable Christmas mouse. Other times he’s written about Star Trek, funny family pet rabbit situations, classic cartoons, and comedy gags. Once a week he shares some kind of humorous statistics.
  4. Solvemymaths shared a great link showing multiple ways to prove the Pythagorean theorem. The 3-4-5 triangle that accompanies his post is far more than a simple illustration. Solvemymaths routinely posts a variety of math problems, gifs, computer programs, and pictures that can help you think and learn to solve your own math problems.
  5. Because I am a vegetarian, I was intrigued by notquitefrenchcuisine‘s old Hungarian family recipe for vegetarian burgers. I also love that she sprinkles a few Hungarian food words in her posts, too.
  6. When I first read Puzzled Over’s Ages-of-three-daughters, I wasn’t sure how the last clue could possibly help me solve it. Maybe it will stump you at first, too, but it really can be solved without too much trouble.
  7. One of the topics Classy Cheapskate blogs about is minimalism. She says a comfortable minimalist owns about 150 things. I can’t imagine that being possible for ANYONE right now, but I do want to head closer to that direction as does she.
  8. I loved Paula Beardell Krieg’s instructions on how to fold paper to make an-equilateral-triangle and other basic shapes. Her method uses the straight edge of a piece of paper, but no compass is needed. Elementary school children can even make perfect squares or triangles using her methods.
  9. Margarita Morris is a young adult author who has shared the complete writing process with her readers. Her books are well researched, written, and revised. She also shares her love of great classical music and can capture a scene in nature without writing a single word.
  10. Resourceaholic scours the internet for resources that teach mathematics. Every one of her posts is loaded with as much information as this link. Your students will learn more if you read her blog, and chances are you will learn something, too!
  11. Nerdinthebrain is a home-schooling parent who loves science and mathematics and sharing her lessons with us. Because of her, my grandson and I had a ton of fun testing the pH of different liquids using red cabbage water. She has MANY good ideas and products.
  12. The Chaos Fairy was also nominated for the Liebster Award at the same time that I was so I’m not counting this as one of my 11. Nevertheless, I just had to recognize the DINOSAUR loving in this blog post.

337 What Will Be the Factors of 2015?

/ ivasallay / Leave a comment

Since this is my 337th post, I’ll first give some information about the number 337, then I will predict the factors for 2015.

  • 337 is a prime number.
  • Prime factorization: 337 is prime.
  • The exponent of prime number 337 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 337 has exactly 2 factors.
  • Factors of 337: 1, 337
  • Factor pairs: 337 = 1 x 337
  • 337 has no square factors that allow its square root to be simplified. √337 ≈ 18.358

How do we know that 337 is a prime number? If 337 were not a prime number, then it would be divisible by at least one prime number less than or equal to √337 ≈ 18.358. Since 337 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 337 is a prime number.

Celebrate the New Year by giving this puzzle a try!

2014-52 Level 3

WHAT will be the FACTORS of the YEAR 2015?

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-29

2014-52 Level 3 Factors

On New Year’s Eve 2013 I predicted that the positive factors for 2014 would be 1, 2, 19, 38, 53, 106,1007, and 2014, and my predictions were 100% accurate!

On this last day of 2014, I boldly announce my predictions for the factors of the year 2015:

  • The positive factors for 2015 will be 1, 5, 13, 31, 65, 155, 403, and 2015
  • Some of these factors will occur in pairs: 1 and 2015, 5 and 403, 13 and 155, as well as 31 and 65.
  • Unfortunately there will be some negative factors in 2015 as well. They will be -1, -5, -13, -31, -65, -155, -403, and -2015.

Whatever life throws your way, I wish you a happy, healthy, and prosperous 2015.

336 and Level 2

/ ivasallay / 2 Comments

Today’s Puzzle:

The sixteen clues given in this puzzle are all you need to complete this multiplication table!

2014-52 Level 2

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-29

2014-52 Level 2 Factors

Factors of 336:

  • 336 is a composite number.
  • Prime factorization: 336 = 2 x 2 x 2 x 2 x 3 x 7, which can be written 336 = (2^4) x 3 x 7
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 x 2 x 2 = 20. Therefore 336 has exactly 20 factors.
  • Factors of 336: 1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 42, 48, 56, 84, 112, 168, 336
  • Factor pairs: 336 = 1 x 336, 2 x 168, 3 x 112, 4 x 84, 6 x 56, 7 x 48, 8 x 42, 12 x 28, 14 x 24 or 16 x 21
  • Taking the factor pair with the largest square number factor, we get √336 = (√16)(√21) = 4√21 ≈ 18.330

Sum-Difference Puzzles:

84 has six factor pairs. One of those factor pairs adds up to 25, and another one subtracts to 25. Can you determine what those factor pairs are to solve the first puzzle below?

336 has ten factor pairs. One of them adds up to 50, and a different one subtracts to 50. 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?

 

335 and Level 1

/ ivasallay / Leave a comment

  • 335 is a composite number.
  • Prime factorization: 335 = 5 x 67
  • 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 335 has exactly 4 factors.
  • Factors of 335: 1, 5, 67, 335
  • Factor pairs: 335 = 1 x 335 or 5 x 67
  • 335 has no square factors that allow its square root to be simplified. √335 ≈ 18.303

If you can multiply, divide and use a little logic, you should easily be able to complete this multiplication table puzzle.

2014-52 Level 1

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-29

2014-52 Level 1 Factors

334 and Level 6

/ ivasallay / Leave a comment

  • 334 is a composite number.
  • Prime factorization: 334 = 2 x 167
  • 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 334 has exactly 4 factors.
  • Factors of 334: 1, 2, 167, 334
  • Factor pairs: 334 = 1 x 334 or 2 x 167
  • 334 has no square factors that allow its square root to be simplified. √334 ≈ 18.276

Solving this Level 6 puzzle can be a little tricky: Is 1 or 2 the common factor for 6 and 8? Is 3 or 6 the common factor for 12 and 30? Is 4 or 8 the common factor for 40 and 16? In each case only one of those choices will make this puzzle work as a multiplication table? Can you figure out what those choices should be? Use logic to find the correct solution, not trial and error.

2014-51 Level 6

Print the puzzles or type the factors on this excel file:  10 Factors 2014-12-22

2014-51 Level 6 Logic

333 and Level 5

/ ivasallay / Leave a comment

  • 333 is a composite number.
  • Prime factorization: 333 = 3 x 3 x 37, which can be written 333 = 3² x 37
  • 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 333 has exactly 6 factors.
  • Factors of 333: 1, 3, 9, 37, 111, 333
  • Factor pairs: 333 = 1 x 333, 3 x 111, or 9 x 37
  • Taking the factor pair with the largest square number factor, we get √333 = (√9)(√37) = 3√37 ≈ 18.248

Can the eleven clues in this puzzle help you to complete this multiplication table?

2014-51 Level 5

Print the puzzles or type the factors on this excel file:  10 Factors 2014-12-22

2014-51 Level 5 Logic