Level 5 Puzzle

316 Christmas Factor Trees

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  • 316 is a composite number.
  • Prime factorization: 316 = 2 x 2 x 79, which can be written 316 = 2² x 79
  • 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 316 has exactly 6 factors.
  • Factors of 316: 1, 2, 4, 79, 158, 316
  • Factor pairs: 316 = 1 x 316, 2 x 158, or 4 x 79
  • Taking the factor pair with the largest square number factor, we get √316 = (√4)(√79) = 2√79 ≈ 17.776

Today is all about Christmas trees. You can read how one family set up a beautiful Christmas tree at three-things-thursday-tree-edition, solve the Christmas tree puzzle, and scroll down to see two possible factor trees for 316.

Then polish up your shoes and boots. Place them near your front door or window so they will be ready to be filled with treats when Mikulás (St. Nickolas) makes his rounds tonight.

2014-48 Level 5

Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-01

316 Christmas Factor Trees

2014-48 Level 5 Logic

 

307 and Level 5

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  • 307 is a prime number.
  • Prime factorization: 307 is prime.
  • The exponent of prime number 307 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 307 has exactly 2 factors.
  • Factors of 307: 1, 307
  • Factor pairs: 307 = 1 x 307
  • 307 has no square factors that allow its square root to be simplified. √307 ≈ 17.521

How do we know that 307 is a prime number? If 307 were not a prime number, then it would be divisible by at least one prime number less than or equal to √307 ≈ 17.521. Since 307 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 307 is a prime number.

Eleven clues that might stump you:

2014-47 Level 5

Print the puzzles or type the factors on this excel file: 10 Factors 2014-11-24

2014-47 Level 5 Logic

301 and Level 5

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  • 301 is a composite number.
  • Prime factorization: 301 = 7 x 43
  • 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 301 has exactly 4 factors.
  • Factors of 301: 1, 7, 43, 301
  • Factor pairs: 301 = 1 x 301 or 7 x 43
  • 301 has no square factors that allow its square root to be simplified. √301 ≈ 17.349

2014-46 Level 5

Print the puzzles or type the factors on this excel file: 12 Factors 2014-11-17

2014-46 Level 5 Logic

294 and Level 5

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Today’s Puzzle:

Can you fill out this multiplication table that has only eleven clues?

2014-45 Level 5

Print the puzzles or type the factors on this excel file: 10 Factors 2014-11-10

2014-45 Level 5 Logic

Factors of 294:

  • 294 is a composite number.
  • Prime factorization: 294 = 2 x 3 x 7 x 7, which can be written 294 = 2 x 3 x (7^2)
  • The exponents in the prime factorization are 1, 1 and 2. Adding one to each and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 x 2 x 3 = 12. Therefore 294 has exactly 12 factors.
  • Factors of 294: 1, 2, 3, 6, 7, 14, 21, 42, 49, 98, 147, 294
  • Factor pairs: 294 = 1 x 294, 2 x 147, 3 x 98, 6 x 49, 7 x 42, or 14 x 21
  • Taking the factor pair with the largest square number factor, we get √294 = (√6)(√49) = 7√6 ≈ 17.146

Sum-Difference Puzzles:

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

294 has six factor pairs. Of of those factor pairs adds up to 35, and a different one subtracts to 35. Identify those factor pairs, and you will be able to solve the second puzzle.

The second puzzle is really just the first puzzle in disguise.

 

288 and Level 5

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  • 288 is a composite number.
  • Prime factorization: 288 = 2 x 2 x 2 x 2 x 2 x 3 x 3, which can be written 288 = (2^5) x (3^2)
  • The exponents in the prime factorization are 5 and 2. Adding one to each and multiplying we get (5 + 1)(2 + 1) = 6 x 3  = 18. Therefore 288 has exactly 18 factors.
  • Factors of 288: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 72, 96, 144, 288
  • Factor pairs: 288 = 1 x 288, 2 x 144, 3 x 96, 4 x 72, 6 x 48, 8 x 36, 9 x 32, 12 x 24, or 16 x 18
  • Taking the factor pair with the largest square number factor, we get √288 = (√2)(√144) = 12√2 ≈ 16.971

2014-44 Level 5

Print the puzzles or type the factors on this excel file: 12 Factors 2014-11-03

2014-44 Level 5 Logic

282 and Level 5

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  • 282 is a composite number.
  • Prime factorization: 282 = 2 x 3 x 47
  • 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 282 has 8 factors.
  • Factors of 282: 1, 2, 3, 6, 47, 94, 141, 282
  • Factor pairs: 282 = 1 x 282, 2 x 141, 3 x 94, or 6 x 47
  • 282 has no square factors that allow its square root to be simplified. √282 ≈ 16.793

Today is Halloween. Give this Level 5 puzzle a try. . . Trick or Treat?

2014-43 Level 5

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-27

2014-43 Level 5 Logic

273 and Level 5

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  • 273 is a composite number.
  • Prime factorization: 273 = 3 × 7 × 13
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2= 8. Therefore 273 has exactly 8 factors.
  • Factors of 273: 1, 3, 7, 13, 21, 39, 91, 273
  • Factor pairs: 273 = 1 × 273, 3 × 91, 7 × 39, or 13 × 21
  • 273 has no square factors that allow its square root to be simplified. √273 ≈ 16.523

Find the Factors is now one of  many  interesting mathematical resources that can be found on Solvemymaths!

2014-42 Level 5

Print the puzzles or type the factors on this excel file: 12 Factors 2014-10-20

2014-42 Level 5 Logic

265 and Level 5

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  • 265 is a composite number.
  • Prime factorization: 265 = 5 x 53
  • 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 265 has 4 factors.
  • Factors of 265: 1, 5, 53, 265
  • Factor pairs: 265 = 1 x 265 or 5 x 53
  • 265 has no square factors that allow its square root to be simplified. √265 ≈ 16.279

Here is today’s puzzle:

2014-41 Level 5

Print the puzzles or type the factors on this excel file: 10 Factors 2014-10-13

2014-41 Level 5 Logic

257 and Level 5

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  • 257 is a prime number.
  • Prime factorization: 257 is prime.
  • The exponent of prime number 257 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 257 has exactly 2 factors.
  • Factors of 257: 1, 257
  • Factor pairs: 257 = 1 x 257
  • 257 has no square factors that allow its square root to be simplified. √257 ≈ 16.031

How do we know that 257 is a prime number? If 257 were not a prime number, then it would be divisible by at least one prime number less than or equal to √257 ≈ 16.031. Since 257 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 257 is a prime number.

257 is 2⁸ + 1 and is the second largest known Fermat Prime.  Click that link to see how unusual and cool Fermat Primes are!

Can you find the factors of the clues in this Level 5 puzzle?

2014-40 Level 5

Print the puzzles or type the factors on this excel file: 12 Factors 2014-10-06

2014-40 Level 5 Logic

250 and Level 5

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  • 250 is a composite number.
  • Prime factorization: 250 = 2 x 5 x 5 x 5, which can be written 250 = 2 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 250 has 8 factors.
  • Factors of 250: 1, 2, 5, 10, 25, 50, 125, 250
  • Factor pairs: 250 = 1 x 250, 2 x 125, 5 x 50, or 10 x 25
  • Taking the factor pair with the largest square number factor, we get √250 = (√10)(√25) = 5√10 ≈ 15.811

2014-39 Level 5

Print the puzzles or type the factors on this excel file: 10 Factors 2014-09-29

2014-39 Level 5 Logic