Math

Prime Number Tests for 347

/ ivasallay / 2 Comments
  • 347 is a prime number.
  • Prime factorization: 347 is prime.
  • The exponent of prime number 347 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 347 has exactly 2 factors.
  • Factors of 347: 1, 347
  • Factor pairs: 347 = 1 x 347
  • 347 has no square factors that allow its square root to be simplified. √347 ≈ 18.628

How do we know that 347 is a prime number? If 347 were not a prime number, then it would be divisible by at least one prime number less than or equal to √347 ≈ 18.628. Since 347 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 347 is a prime number.

Is there a quicker way to tell that 347 is a prime number? Not exactly…There is a quick test that will tell you if 347 is VERY LIKELY prime.

Quick Prime or Composite Number Test: You can use as few as TEN calculator keystrokes to test if 347 is VERY LIKELY prime: What is the remainder when 2³⁴⁷ is divided by 347? To find out type “(2)(xʸ)(3)(4)(7)(Mod)(3)(4)(7)” followed by the equal sign into your computer’s calculator. This is how the calculator should look before you hit the equal sign:

mod 347

2^347 mod (347) = 2, the same as the base we typed in. That means 347 is VERY LIKELY prime. This tests works for all prime numbers, but sometimes it gives a false positive for a relatively few (but infinite number) of composite numbers. (341 is the smallest of these numbers, and I wrote about it here.)

2 isn’t the only prime number that can be used as the base in this quick test. You can verify the following also on your computer’s calculator:

  • 3^347 mod (347) = 3
  • 5^347 mod (347) = 5
  • 7^347 mod (347) = 7
  • 11^347 mod (347) = 11. We can practically go on forever using prime number after prime number….until
  • 337^347 mod (347) = 337, the largest prime number less than 347.

As long as we use a prime less than 347 as the base and 347 as the exponent, we will always get that prime as the remainder. Still even though 347 passes ALL those prime number tests, we can only conclude that 347 is VERY LIKELY prime. Doing all those calculations is more work than simply dividing 347 by all the prime numbers less than or equal to its square root and getting a remainder every single time.

Tomcircle, a fellow blogger, shared a video of a new-prime-number-test that works for prime numbers and NEVER gives a false positive.

While this test works every time in theory, it can be quite a nightmare in practice. It involves putting the possible prime number into a particular expression, expanding the expression, calculating each coefficient, and verifying that all those coefficients are divisible by this possible prime number. In the case of the relatively small number 347, there would be 173 different coefficients. Each of those coefficients are the numbers in the 347th row of Pascal’s triangle, and too many of them are usually expressed in scientific notation. Dividing each of them by 347 to verify there is no remainder is far more work than most people would want to do. In fact, dividing 347 by all the integers less than 347 would actually be less work!

The Sieve of Eratosthenes was how the ancient Greeks found prime numbers. It takes a little longer than dividing the possible prime number by all of the primes less than or equal to its square root, but it finds many primes at the same time. Solvemymath shares some interesting facts about it.

Prime numbers are intriguing, and 347 is one of them!

346 and Level 4

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Quick Prime/Composite Number Test: 346 is even and thus divisible by 2, so 346 is a composite number.

346 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-01-05

  • 346 is a composite number.
  • Prime factorization: 346 = 2 x 173
  • 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 346 has exactly 4 factors.
  • Factors of 346: 1, 2, 173, 346
  • Factor pairs: 346 = 1 x 346 or 2 x 173
  • 346 has no square factors that allow its square root to be simplified. √346 ≈ 18.601

346 Logic

345 and Level 3

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Quick Prime or Composite Number Test: The last digit of 345 is 5, so it can be evenly divided by 5. Therefore, 345 is a composite number.

345 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-01-05

  • 345 is a composite number.
  • Prime factorization: 345 = 3 x 5 x 23
  • 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 345 has exactly 8 factors.
  • Factors of 345: 1, 3, 5, 15, 23, 69, 115, 345
  • Factor pairs: 345 = 1 x 345, 3 x 115, 5 x 69, or 15 x 23
  • 345 has no square factors that allow its square root to be simplified. √345 ≈ 18.574

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.

345 Factors

344 and Level 2

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Quick Prime or Composite Number Test: 344 is even and thus divisible by 2, so 344 is a composite number.

344 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-01-05

  • 344 is a composite number.
  • Prime factorization: 344 = 2 x 2 x 2 x 43, which can be written (2^3) x 43
  • 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 344 has exactly 8 factors.
  • Factors of 344: 1, 2, 4, 8, 43, 86, 172, 344
  • Factor pairs: 344 = 1 x 344, 2 x 172, 4 x 86, or 8 x 43
  • Taking the factor pair with the largest square number factor, we get √344 = (√4)(√86) = 2√86 ≈ 18.547

344 Factors

343 and a Few Changes

/ ivasallay / 6 Comments

Starting today, I’ve made a few small changes to the way my posts will look. If you can spot what the changes are, let me know if you have an opinion about them.

Quick Prime or Composite Number Test: If you divide 2^343 by 343, the remainder is 324, not 2, so 343 is NOT a prime number and is a composite number. (Use the mod function on the computer’s calculator to find the remainder.) Scroll down to see 343’s factors.

343 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-01-05

  • 343 is a composite number.
  • Prime factorization: 343 = 7 x 7 x 7, which can be written 343 = 7³
  • The exponent in the prime factorization is 3. Adding one, we get (3 + 1) = 4. Therefore 343 has exactly 4 factors.
  • Factors of 343: 1, 7, 49, 343
  • Factor pairs: 343 = 1 x 343 or 7 x 49
  • Taking the factor pair with the largest square number factor, we get √343 = (√7)(√49) = 7√7 ≈ 18.520
  • 343 is a perfect cube.

OEIS.org tells us that (3+4)³ = 343, so 343 is a Friedman number, a number that can be expressed using only its own digits and (), +, -, ×, and ÷.

343 Factors

342 and Level 6

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  • 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

340 and Level 5

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  • 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

337 What Will Be the Factors of 2015?

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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

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  • 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