325 is a Triangular Number

/ ivasallay / 3 Comments

  • 325 is a composite number.
  • Prime factorization: 325 = 5 x 5 x 13, which can be written 325 = 5² x 13
  • 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 325 has exactly 6 factors.
  • Factors of 325: 1, 5, 13, 25, 65, 325
  • Factor pairs: 325 = 1 x 325, 5 x 65, or 13 x 25
  • Taking the factor pair with the largest square number factor, we get √325 = (√13)(√25) = 5√13 ≈ 18.028

325 is a triangular number. 1 + 2 + 3 + 4 + . . . + 22 + 23 + 24 + 25 = 325. Shorthand for that sum of 25 numbers is given with the ∑ sign in the graphic below. The way to find the sum quickly using multiplication is also given:

325 - Triangular Number

If you look at this Level 3 puzzle from far enough away, you also might see a triangle!

2014-50 Level 3

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

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.

2014-50 Level 3 Factors

324 Christmas Factor Trees

/ ivasallay / 2 Comments

  • 324 is a composite number.
  • Prime factorization: 324 = 2 x 2 x 3 x 3 x 3 x 3, which can be written 324 = (2^2) x (3^4)
  • The exponents in the prime factorization are 2 and 4. Adding one to each and multiplying we get (2 + 1)(4 + 1) = 3 x 5 = 15. Therefore 324 has exactly 15 factors.
  • Factors of 324: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, 162, 324
  • Factor pairs: 324 = 1 x 324, 2 x 162, 3 x 108, 4 x 81, 6 x 54, 9 x 36, 12 x 27, or 18 x 18
  • 324 is a perfect square. √324 = 18

Included in these 324 factor trees are smaller factor trees for the numbers in brown: 4, 6, 9, 12, 18, 27, 36, 54, 81, 108, and 162. Prime factors of 324 are in red.

324 Factor Trees

This Level 2 puzzle looks like part of a Christmas tree branch. It’s not too difficult to solve. Give it a try!

2014-50 Level 2

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

2014-50 Level 2 Factors

 

323 and Level 1

/ ivasallay / Leave a comment

  • 323 is a composite number.
  • Prime factorization: 323 = 17 x 19
  • 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 323 has exactly 4 factors.
  • Factors of 323: 1, 17, 19, 323
  • Factor pairs: 323 = 1 x 323 or 17 x 19
  • 323 has no square factors that allow its square root to be simplified. √323 ≈ 17.972

Since 17 × 19 = 323, we know that 323 + 1 = 324 = 18². That follows from the fact that (n – 1)(n + 1) = n² – 1 is ALWAYS true.

Because 17 is one of its factors, 323 is the hypotenuse of a Pythagorean triple:

  • 152-285-323 which is 19 times 8-15-17

Here are 20 very easy clues to help you solve this puzzle:

2014-50 Level 1

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

2014-50 Level 1 Factors

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

321 I Heard the Bells on Christmas Day

/ ivasallay / 2 Comments

321 is made solely from three consecutive numbers so it can be evenly divided by 3.

  • 321 is a composite number.
  • Prime factorization: 321 = 3 x 107
  • 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 321 has exactly 4 factors.
  • Factors of 321: 1, 3, 107, 321
  • Factor pairs: 321 = 1 x 321 or 3 x 107
  • 321 has no square factors that allow its square root to be simplified. √321 ≈ 17.916
I heard the bells
Print the puzzles or type the factors on this excel file: 10 Factors 2014-12-08

One Christmas day about the time of the Civil War, Henry Wadsworth Longfellow wrote a poem he titled “Christmas Bells”. 5/7 of that poem later became the song known as “I Heard the Bells on Christmas Day”. You can read the poem’s complete original text here.

I heard the bells factors

320 Christmas Factor Trees

/ ivasallay / 1 Comment

  • 320 is a composite number.
  • Prime factorization: 320 = 2 x 2 x 2 x 2 x 2 x 2 x 5, which can be written 320 = 2⁶ x 5
  • The exponents in the prime factorization are 6, and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1) = 7 x 2 = 14. Therefore 320 has exactly 14 factors.
  • Factors of 320: 1, 2, 4, 5, 8, 10, 16, 20, 32, 40, 64, 80, 160, 320
  • Factor pairs: 320 = 1 x 320, 2 x 160, 4 x 80, 5 x 64, 8 x 40, 10 x 32, or 16 x 20
  • Taking the factor pair with the largest square number factor, we get √320 = (√5)(√64) = 8√5 ≈ 17.889

In these factor trees for 320, we can also see factor trees for 4, 8, 10, 16, 20, 32, 40, 64, 80, and 160. The tops of each of those factor trees are in brown. No matter which factor pairs we use to build the tree, we always get the same prime factors which are indicated in red.

320 Factor Trees

Now try finding the factors in this Christmas tree puzzle that make it work like a multiplication table:

2014-49 Level 3

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

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.

2014-49 Level 3 Factors

 

319 and Level 2

/ ivasallay / Leave a comment

  • 319 is a composite number.
  • Prime factorization: 319 = 11 x 29
  • 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 319 has exactly 4 factors.
  • Factors of 319: 1, 11, 29, 319
  • Factor pairs: 319 = 1 x 319 or 11 x 29
  • 319 has no square factors that allow its square root to be simplified. √319 ≈ 17.861

Since 29 is one of its factors, 319 is the hypotenuse of a Pythagorean triple:

  • 220-231-319 which is 11 times 20-21-29

This Level 2 puzzle has more than enough clues to help you find all of its factors:

2014-49 Level 2

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

2014-49 Level 2 Factors

318 Christmas Factor Tree Gif

/ ivasallay / 2 Comments

  • 318 is a composite number.
  • Prime factorization: 318 = 2 x 3 x 53
  • 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 318 has exactly 8 factors.
  • Factors of 318: 1, 2, 3, 6, 53, 106, 159, 318
  • Factor pairs: 318 = 1 x 318, 2 x 159, 3 x 106, or 6 x 53
  • 318 has no square factors that allow its square root to be simplified. √318 ≈ 17.833

This factoring tree gif solves and completes this Level ONE puzzle right before your eyes. Scroll down to see three possible factor trees for the number 318.

Christmas tree factor puzzle on Make A Gif

make animated gifs like this at MakeAGif

318 Christmas Factor Trees

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

 

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.

316 Christmas Factor Trees

/ ivasallay / Leave a comment

  • 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