factor pairs

332 and a Christmas Star

/ ivasallay / Leave a comment

  • 332 is a composite number.
  • Prime factorization: 332 = 2 x 2 x 83 which can be written 2² x 83
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2 = 6. Therefore 332 has exactly 6 factors.
  • Factors of 332: 1, 2, 4, 83, 166, 332
  • Factor pairs: 332 = 1 x 332, 2 x 166, or 4 x 83
  • Taking the factor pair with the largest square number factor, we get √332 = (√4)(√83) = 2√83 ≈ 18.221

Merry Christmas! This is a rather easy level 5 puzzle so I’m sharing it instead of a level 4 puzzle today.

2014-51 Level 4

Print the puzzles or type the factors on this excel file:  10 Factors 2014-12-22

The Night Wind excitedly asked a little lamb and the whole world, “Do you see what I see?” when it saw “a star, a star, dancing in the night” after the Christ child was born. Click here to read the complete lyrics of “Do You Hear What I Hear.”

2014-51 Level 4 Factors

331 and Hockey Sticks

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  • 331 is a prime number.
  • Prime factorization: 331 is prime.
  • The exponent of prime number 331 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 331 has exactly 2 factors.
  • Factors of 331: 1, 331
  • Factor pairs: 331 = 1 x 331
  • 331 has no square factors that allow its square root to be simplified. √331 ≈ 18.193

How do we know that 331 is a prime number? If 331 were not a prime number, then it would be divisible by at least one prime number less than or equal to √331 ≈ 18.193. Since 331 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 331 is a prime number.

2014-51 Level 3

 Print the puzzles or type the factors on this excel file:  10 Factors 2014-12-22

Today’s puzzle looks like a hockey stick. Hockey sticks remind me not only of the obvious winter sport but also of the Twelve Days of Christmas and Pascal’s triangle.

Dimacs.rutgers.edu explains quite nicely how a hockey stick in Pascal’s triangle can give you the total number of gifts received after one day, two days, three days, and so on. Look at the green and red hockey stick with bold black numbers in this illustration of Pascal’s triangle:

If someone gave you one partridge every day for 12 days, two turtle doves every day for 11 days, three French hens every day for 10 days, etc, etc, and etc, then you would receive 364 gifts. (364 is so easy to remember because it is one less than the number of days in a year.)

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.
2014-51 Level 3 Factors

330 Christmas Factor Trees

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Today’s Puzzle:

Can you find the factors and complete this Christmas tree multiplication table?2014-51 Level 2

Print the puzzles or type the factors on this excel file:  10 Factors 2014-12-22

Factor Trees for 330:

Within these seven factor trees for 330 there are also factor trees for 6, 10, 15, 22, 30, 33, 55, 66, 110, and 165, the tops of which are all in brown. The prime factors of 330 are all in red.

330 Factor Trees

Factors of 330:

  • 330 is a composite number.
  • Prime factorization: 330 = 2 x 3 x 5 x 11
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 330 has exactly 16 factors.
  • Factors of 330: 1, 2, 3, 5, 6, 10, 11, 15, 22, 30, 33, 55, 66, 110, 165, 330
  • Factor pairs: 330 = 1 x 330, 2 x 165, 3 x 110, 5 x 66, 6 x 55, 10 x 33, 11 x 30, or 15 x 22
  • 330 has no square factors that allow its square root to be simplified. √330 ≈ 18.166

Sum-Difference Puzzle:

330 has eight factor pairs. The numbers in one of those pairs add up to 61, and the numbers in another one subtract to 61. If you can identify those factors, then you can solve this puzzle!

Tree Puzzle Solution:

2014-51 Level 2 Factors

 

 

 

329 and A Last Minute Gift

/ ivasallay / Leave a comment

  • 329 is a composite number.
  • Prime factorization: 329 = 7 x 47
  • 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 329 has exactly 4 factors.
  • Factors of 329: 1, 7, 47, 329
  • Factor pairs: 329 = 1 x 329 or 7 x 47
  • 329 has no square factors that allow its square root to be simplified. √329 ≈ 18.138

Here’s a puzzle that could be a last minute gift. If there is a child in you life who has recently become familiar with the multiplication table, put this puzzle in his or her stocking! It is as easy as the puzzles get.

2014-51 Level 1

Print or type on this week’s puzzles using this excel file: 10 Factors 2014-12-22

2014-51 Level 1 Factors

Last year on Christmas Eve I offered a free last minute gift, a puzzle booklet, but I discovered that most people take care of their last minute gift-giving a couple of days sooner than that. Still this last minute gift could be sent electronically and you can’t beat the price.

I also learned that margins in excel can move so the booklet didn’t always print up as nicely as I wanted. I’m still recovering from surgery so this year I simply revised that same booklet.

I saved the booklet as Factor Holiday pdf to eliminate those printing issues.  In pdf, the lines on the puzzles became quite dark so I don’t like the way they look as much, but I’ll live with it.

The booklet is also available in Factor Holiday excel if you prefer to type your answers directly on the computer. You can try printing it off of excel as well, but I can’t guarantee what the margins will do at any given time.

Here’s a copy of the puzzle booklet’s cover with my sincere holiday greetings for all of you:

2013 Puzzle Holiday

328 and Level 6

/ ivasallay / 1 Comment

  • 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

327 and Level 5

/ ivasallay / Leave a comment

  • 327 is a composite number.
  • Prime factorization: 327 = 3 x 109
  • 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 327 has exactly 4 factors.
  • Factors of 327: 1, 3, 109, 327
  • Factor pairs: 327 = 1 x 327 or 3 x 109
  • 327 has no square factors that allow its square root to be simplified. √327 ≈ 18.083

Can these 14 clues help you complete this multiplication table puzzle?

2014-50 Level 5

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

2014-50 Level 5 Logic

326 Tiny Christmas Factor Tree

/ ivasallay / Leave a comment

  • 326 is a composite number.
  • Prime factorization: 326 = 2 x 163
  • 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 326 has exactly 4 factors.
  • Factors of 326: 1, 2, 163, 326
  • Factor pairs: 326 = 1 x 326 or 2 x 163
  • 326 has no square factors that allow its square root to be simplified. √326 ≈ 18.055

Even though 326 is a three digit number, there is only one way to construct its factor tree, two if you count its mirror image. Either way is illustrated here. Below them is a Christmas factor tree puzzle that is a lot more interesting than the factor tree for 326.

2014-50 Level 4

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

2014-50 Level 4 Logic

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