number puzzle

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

/ 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

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

339 and Level 4

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  • 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

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

/ 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

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  • 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

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  • 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