Level 6 Puzzle

401 and Level 6

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We can easily see that 401 is not divisible by 2, 3, or 5. Is it a prime number? Let’s do a quick test: 2^401 (mod 401) = 2, so 401 is VERY LIKELY a prime number. Scroll down past the puzzle to know for sure.

401 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-16

  • 401 is a prime number.
  • Prime factorization: 401 is prime.
  • The exponent of prime number 401 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 401 has exactly 2 factors.
  • Factors of 401: 1, 401
  • Factor pairs: 401 = 1 x 401
  • 401 has no square factors that allow its square root to be simplified. √401 ≈ 20.025

How do we know that 401 is a prime number? If 401 were not a prime number, then it would be divisible by at least one prime number less than or equal to √401 ≈ 20.025. Since 401 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 401 is a prime number.

Pythagorean triples: Since 401 is a prime number, it can only be a part of primitive Pythagorean triples: 40-399-401 and 401-80400-80401.

401 Logic

392 Happy Valentine’s Day

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392 is divisible by seven because 39 minus two times the last digit is 35, a multiple of seven. That is one of the reasons 392 is a composite number. All of its factors are listed below the puzzle.

392 Puzzle

Print the puzzles or type the factors on this excel file:12 Factors 2015-02-09

  • 392 is a composite number.
  • Prime factorization: 392 = 2 x 2 x 2 x 7 x 7, which can be written 392 = (2^3) x (7^2)
  • The exponents in the prime factorization are 3 and 2. Adding one to each and multiplying we get (3 + 1)(2 + 1) = 4 x 3 = 12. Therefore 392 has exactly 12 factors.
  • Factors of 392: 1, 2, 4, 7, 8, 14, 28, 49, 56, 98, 196, 392
  • Factor pairs: 392 = 1 x 392, 2 x 196, 4 x 98, 7 x 56, 8 x 49, or 14 x 28
  • Taking the factor pair with the largest square number factor, we get √392 = (√2)(√196) = 14√2 ≈ 19.799

392 Factors

384 and Level 6

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Today’s Puzzle:

384 Puzzle

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

Here is a logical way to use the clues in the puzzle to find the solution:

384 Logic

Factors of 384:

384 is even and, therefore, composite.

  • 384 is a composite number.
  • Prime factorization: 384 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3, which can be written 384 = (2^7) x 3
  • The exponents in the prime factorization are 7 and 1. Adding one to each and multiplying we get (7 + 1)(1 + 1) = 8 x 2 = 16. Therefore 384 has exactly 16 factors.
  • Factors of 384: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 384
  • Factor pairs: 384 = 1 x 384, 2 x 192, 3 x 128, 4 x 96, 6 x 64, 8 x 48, 12 x 32, or 16 x 24
  • Taking the factor pair with the largest square number factor, we get √384 = (√6)(√64) = 8√6 ≈ 18.596

Sum-Difference Puzzles:

6 has two factor pairs. One of those pairs adds up to 5, and the other one subtracts to 5. Put the factors in the appropriate boxes in the first puzzle.

384 has eight factor pairs. One of the factor pairs adds up to 40, and a different one subtracts to 40. 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?

More about the Number 384:

384 is in this cool pattern:

 

375 and Level 6

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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

365 and Level 6

/ ivasallay / 2 Comments

If the earth revolved around the sun just a little bit faster there would be 364 days in a year, and those 364 days could easily be divided into 52-seven day weeks, thirteen 28-day months, and four 91-day seasons. If you were born on a Saturday, your birthday would occur on a Saturday every single year, too. Would there be more superstitions about the number 13 or less?

But that isn’t the way things are. It takes the earth nearly 365 1/4 days to revolve around the sun. We have settled on 365 days in a year with a 366-day leap year almost every four years.

365 has only two prime factors: 5 and 73. Seventy-three is a much bigger prime number than people use regularly, but it is only one more than 72 which has lots of great factors including twelve, a number that can easily be divided in half or into four seasons.

Besides if we could make the earth revolve around the sun faster, why not choose 360 days instead? Then we could have twelve 30-day months, four 90-day seasons, and our choice of 60 six-day weeks or 45 eight-day weeks. Perhaps we could listen to the Beatles sing Eight Days a Week.

On the other hand, I suppose we would all grow a little older a little faster…oops! Just forget I brought up the subject, please! And do something to keep your brain young, like this puzzle:

365 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-01-19

  • 365 is a composite number.
  • Prime factorization: 365 = 5 x 73
  • 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 365 has exactly 4 factors.
  • Factors of 365: 1, 5, 73, 365
  • Factor pairs: 365 = 1 x 365 or 5 x 73
  • 365 has no square factors that allow its square root to be simplified. √365 ≈ 19.105

365 Logic

356 and Level 6

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356 is even so it is a composite number. Scroll down past the puzzle to see its factors.

356 Puzzle

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

  • 356 is a composite number.
  • Prime factorization: 356 = 2 x 2 x 89, which can be written 356 = (2^2) x 89
  • 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 356 has exactly 6 factors.
  • Factors of 356: 1, 2, 4, 89, 178, 356
  • Factor pairs: 356 = 1 x 356, 2 x 178, or 4 x 89
  • Taking the factor pair with the largest square number factor, we get √356 = (√4)(√89) = 2√89 ≈ 18.868

356 Logic

349 and Level 6

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When 2^349 is divided by 349, the remainder is 2, so 349 is VERY LIKELY a prime number. Scroll down past the puzzle to know for sure.

349 Puzzle

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

  • 349 is a prime number.
  • Prime factorization: 349 is prime.
  • The exponent of prime number 349 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 349 has exactly 2 factors.
  • Factors of 349: 1, 349
  • Factor pairs: 349 = 1 x 349
  • 349 has no square factors that allow its square root to be simplified. √349 ≈ 18.682

How do we know that 349 is a prime number? If 349 were not a prime number, then it would be divisible by at least one prime number less than or equal to √349 ≈ 18.682. Since 349 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 349 is a prime number.

Prime Number Trivia: There are sixteen prime numbers between 300 and 400. Of those, 349 is the only one that doesn’t produce another prime number when 300 is subtracted from it. 349 – 300 = 49, a composite number. This piece of trivia might be helpful if you wanted to memorize all the prime numbers less than 1000.

349 Logic

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

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

328 and Level 6

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  • 328 is a composite number.
  • Prime factorization: 328 = 2 x 2 x 2 x 41, which can be written (2^3) x 41
  • 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 328 has exactly 8 factors.
  • Factors of 328: 1, 2, 4, 8, 41, 82, 164, 328
  • Factor pairs: 328 = 1 x 328, 2 x 164, 4 x 82, or 8 x 41
  • Taking the factor pair with the largest square number factor, we get √328 = (√4)(√82) = 2√82 ≈ 18.111

Will these 13 clues in this puzzle stump you?

2014-50 Level 6

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-15

2014-50 Level 6 Logic