Level 6 Puzzle

322 and a Note of Music

/ ivasallay / 2 Comments

  • 322 is a composite number.
  • Prime factorization: 322 = 2 x 7 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 322 has exactly 8 factors.
  • Factors of 322: 1, 2, 7, 14, 23, 46, 161, 322
  • Factor pairs: 322 = 1 x 322, 2 x 161, 7 x 46, or 14 x 23
  • 322 has no square factors that allow its square root to be simplified. √322 ≈ 17.944

I have enjoyed reading what some of my fellow bloggers have written about Christmas music. I highly recommend the following posts:

  1. your-christmas-songs-schedule
  2. i-has-the-feels-a-christmas-song
  3. german-lieder-winterreise-gute-nacht
  4. how-soon-is-too-soon-for-holiday-music?
  5. the-most-wonderful-song-of-the-year which included this fabulous interactive graphic of time magazine’s list of the most popular christmas-songs.

Beautiful music can be a challenge to produce. This Level 6 puzzle in the shape of a musical note can also be challenging, but it can be completed using logic:

2014-49 Level 6

Print the puzzles or type the factors on this excel file: 10 Factors 2014-12-08

2014-49 Level 6 Logic

317 and Candies from Mikulás

/ ivasallay / Leave a comment

  • 317 is a prime number.
  • Prime factorization: 317 is prime.
  • The exponent of prime number 317 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 317 has exactly 2 factors.
  • Factors of 317: 1, 317
  • Factor pairs: 317 = 1 x 317
  • 317 has no square factors that allow its square root to be simplified. √317 ≈ 17.804

How do we know that 317 is a prime number? If 317 were not a prime number, then it would be divisible by at least one prime number less than or equal to √317 ≈ 17.804. Since 317 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 317 is a prime number.

Today is Mikulás or St. Nickolas’s name day in Hungary. Children woke up to find various candies such as the candy cane below in their shined shoes this morning.

2014-48 Level 6

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

2014-48 Level 6 Logic

My grandson left this note for Mikulás last night.

“Köszönöm Mikulás” means “Thank you, St. Nickolas.” My grandson was thrilled with the candies and amused by the virgács that he found in his shoes this Mikulás morning.

309 and a Birthday Cake

/ ivasallay / Leave a comment

  • 309 is a composite number.
  • Prime factorization: 309 = 3 x 103
  • 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 309 has exactly 4 factors.
  • Factors of 309: 1, 3, 103, 309
  • Factor pairs: 309 = 1 x 309 or 3 x 103
  • 309 has no square factors that allow its square root to be simplified. √309 ≈ 17.578

Happy birthday to my brother, Andy. Enjoy this level 6 cake! Love, Iva

2014 birthday Andy

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

2014 birthday Andy Logic

308 and Level 6

/ ivasallay / Leave a comment

  • 308 is a composite number.
  • Prime factorization: 308 = 2 × 2 × 7 × 11, which can be written 308 = 2² × 7 × 11
  • 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 308 has exactly 12 factors.
  • Factors of 308: 1, 2, 4, 7, 11, 14, 22, 28, 44, 77, 154, 308
  • Factor pairs: 308 = 1 x 308, 2 x 154, 4 x 77, 7 x 44, 11 x 28, or 14 x 22
  • Taking the factor pair with the largest square number factor, we get √308 = (√4)(√77) = 2√77 ≈ 17.5499

2014-47 Level 6

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

2014-47 Level 6 Logic

302 and Level 6

/ ivasallay / Leave a comment

  • 302 is a composite number.
  • Prime factorization: 302 = 2 x 151
  • 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 302 has exactly 4 factors.
  • Factors of 302: 1, 2, 151, 302
  • Factor pairs: 302 = 1 x 302 or 2 x 151
  • 302 has no square factors that allow its square root to be simplified. √302 ≈ 17.378

Here’s a very tricky level 6 puzzle to try.

2014-46 Level 6

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

The logic for this one is complicated. Here is an explanation that may be helpful:

9 is a clue two times in this puzzle. At most one of those 9’s can be 3 x 3. Clue 99 and the other nine will use both 9’s so 72 cannot be 8 x 9 in this puzzle. Here is an explanation of why 72 must be 6 x 12 with the 6 in the first column and the 12 in the top row.

Assuming 6 goes over 72 in top row

Now we can find the rest of the factors easily using logic:

2014-46 Level 6 Logic

 

295 and Level 6

/ ivasallay / Leave a comment

  • 295 is a composite number.
  • Prime factorization: 295 = 5 x 59
  • 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 295 has exactly 4 factors.
  • Factors of 295: 1, 5, 59, 295
  • Factor pairs: 295 = 1 x 295 or 5 x 59
  • 295 has no square factors that allow its square root to be simplified. √295 ≈ 17.176

These eleven clues could be pretty tricky!

2014-45 Level 6

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

2014-45 Level 6 Logic

289 and Level 6

/ ivasallay / Leave a comment

  • 289 is a composite number.
  • Prime factorization: 289 = 17 x 17, which can be written 289 = (17^2)
  • The exponent in the prime factorization is 2. Adding one we get (2 + 1) =  3. Therefore 289 has exactly 3 factors.
  • Factors of 289: 1, 17, 289
  • Factor pairs: 289 = 1 x 289 or 17 x 17
  • 289 is a perfect square. √289 = 17

1 + 3 + 5 + 7 + 9 + 11 + … + 29 + 31 + 33 = 289

Teachfurthermaths is now including Find the Factors in its list of useful links!

2014-44 Level 6

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

2014-44 Level 6 Logic

283 and Level 6

/ ivasallay / Leave a comment

  • 283 is a prime number.
  • Prime factorization: 283 is prime.
  • The exponent of prime number 283 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 283 has exactly 2 factors.
  • Factors of 283: 1, 283
  • Factor pairs: 283 = 1 x 283
  • 283 has no square factors that allow its square root to be simplified. √283 ≈ 16.823

How do we know that 283 is a prime number? If 283 were not a prime number, then it would be divisible by at least one prime number less than or equal to √283 ≈ 16.823. Since 283 cannot be divided evenly by 2, 3, 5, 7, 11, or 13, we know that 283 is a prime number.

2014-43 Level 6

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

2014-43 Level 6 Logic

274 and Level 6

/ ivasallay / 2 Comments

  • 274 is a composite number.
  • Prime factorization: 274 = 2 x 137
  • 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 274 has 4 factors.
  • Factors of 274: 1, 2, 137, 274
  • Factor pairs: 274 = 1 x 274 or 2 x 137
  • 274 has no square factors that allow its square root to be simplified. √274 ≈ 16.553.

I recommend using logic and not guess and check to solve the Find the Factors puzzles, but the logic for the one below can be a little tricky. If necessary, scroll down for one piece of reasoning that may be helpful if you get stuck in the middle of trying to solve this one.

2014-42 Level 6

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

Notes on some logic needed for this puzzle:

  • The choices for 24 are 2 x 12, 3 x 8, or 4 x 6.
  • Because 28 = 4 x 7, 16 = 2 x 8, not 4 x 4. Thus 16 will use one of the 2’s. Either 6 or 8 will use the other 2, so we eliminate  2 x 12 = 24.
  • Also 21 will use one of the 3’s while 6 or 18 will use the other. Thus we eliminate  3 x 8 = 24.
  • Therefore 24 = 4 x 6 or 6 x 4.

2014-42 Level 6 Logic

 

267 and Level 6

/ ivasallay / Leave a comment

  • 267 is a composite number.
  • Prime factorization: 267 = 3 x 89
  • 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 267 has 4 factors.
  • Factors of 267: 1, 3, 89, 267
  • Factor pairs: 267 = 1 x 267 or 3 x 89
  • 267 has no square factors that allow its square root to be simplified. √267 ≈ 16.3401

Can you solve this level 6 puzzle? Possible steps to logically arrive at the solution are given at the end of the post.

2014-41 Level 6

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

2014-41 Level 6 Logic