factor

315 Christmas Factor Trees

/ ivasallay / 1 Comment

315 is the smallest number that is divisible by every 1-digit odd number.

  • 315 is a composite number.
  • Prime factorization: 315 = 3 × 3 × 5 × 7, which can be written 315 = 3² × 5 × 7
  • 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 315 has exactly 12 factors.
  • Factors of 315: 1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315
  • Factor pairs: 315 = 1 x 315, 3 x 105, 5 x 63, 7 x 45, 9 x 35, or 15 x 21
  • Taking the factor pair with the largest square number factor, we get √315 = (√9)(√35) = 3√35 ≈ 17.748

Five of the six factor pairs for 315 make lovely Christmas factor trees with the prime factors in red:

315 Christmas Factor Trees

Here’s a level 4 puzzle that is different kind of Christmas factor tree. Can you complete this mixed-up multiplication table when you’re given only these twelve clues?

2014-48 Level 4

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

2014-48 Level 4 Logic

314 and The Pi Day of Our Lives

/ ivasallay / 2 Comments

  • 314 is a composite number.
  • Prime factorization: 314 = 2 x 157
  • 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 314 has exactly 4 factors.
  • Factors of 314: 1, 2, 157, 314
  • Factor pairs: 314 = 1 x 314 or 2 x 157
  • 314 has no square factors that allow its square root to be simplified. √314 ≈ 17.720

On 3-14-15 at 9:26:53 we will experience the most significant pi day of our lives.

Thepidayofourlives.homestead.com  is selling wristbands, t-shirts, and baseball caps commemorating this special upcoming event at very reasonable prices.

Each wristband costs only $3.14 (minimum purchase is 3 wristbands) while the t-shirts and the baseball caps are only $20 each. The website offers free shipping on everything they offer anywhere! You might just want to add one or more of these items to your Christmas wishlist!

 

 

313 and Memorizing One Seventh

/ ivasallay / 1 Comment

  • 313 is a prime number.
  • Prime factorization: 313 is prime.
  • The exponent of prime number 313 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 313 has exactly 2 factors.
  • Factors of 313: 1, 313
  • Factor pairs: 313 = 1 x 313
  • 313 has no square factors that allow its square root to be simplified. √313 ≈ 17.692

How do we know that 313 is a prime number? If 313 were not a prime number, then it would be divisible by at least one prime number less than or equal to √313 ≈ 17.692. Since 313 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 313 is a prime number.

Question: Liam set a goal to read all of the Harry Potter books. What percentage of the books had he read once he finished the first book? Answer: 1/7 ≈ 14.2857%, which looks like a much more complicated percentage than it really is. 1/7 is a repeating decimal that isn’t too difficult to remember. Here’s how I remember it:

one seventh

A famous approximation for pi is 22/7 = 3 1/7. If you have 1/7 memorized then you can also easily remember that pi is approximately equal to 3.142857. (It’s true that 355/113 ≈ 3.141592 is a better approximation since it’s accurate to six decimal places while 22/7 is accurate to only two. You can remember 355/113 by thinking 113355 and putting the first three digits in the denominator and the last three digits in the numerator.)

Here’s some other questions to consider: What percentage of the names of the seven dwarfs can you remember? What decimal amount of the seven wonders of the world can you name? What percentage of days last week did you exercise? How many of the seven deadly sins….

The decimals for 2/7, 3/7, 4/7, 5/7, and 6/7 are also easy and use the same very cool pattern, 142857, but each fraction starts with a different digit. Here’s how to remember each of those decimals, too:

remembering the sevenths

Go ahead and make somebody think you’re a human calculator today!

 

312 and Virgács

/ ivasallay / 1 Comment

Today might not be the 312th day of the year, but here’s a fun fact about the number 312:

//platform.twitter.com/widgets.js

  • 312 is a composite number.
  • Prime factorization: 312 = 2 x 2 x 2 x 3 x 13, which can be written 312 = (2^3) x 3 x 13
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 312 has exactly 16 factors.
  • Factors of 312: 1, 2, 3, 4, 6, 8, 12, 13, 24, 26, 39, 52, 78, 104, 156, 312
  • Factor pairs: 312 = 1 x 312, 2 x 156, 3 x 104, 4 x 78, 6 x 52, 8 x 39, 12 x 26, or 13 x 24
  • Taking the factor pair with the largest square number factor, we get √312 = (√4)(√78) = 2√78 ≈ 17.664

The night of December 5th, Mikulás, or St. Nickolas, will visit homes in Hungary, some neighboring countries, and even a few houses in the United States.  Children who have been good will awaken to find their shoes filled with little treats such as candy, fruit or nuts. Since all children are occasionally a little bit naughty, they will also find virgács, a small collection of twigs that have been spray-painted gold and decoratively bound together.  Virgács are not sold in the United States, so St Nickolas will be making some himself using the bristles from a natural broom.  The finished product should look like today’s puzzle.

This puzzle is meant to be reminiscent of the virgács my children’s ancestors probably received each year on St Nickolas Day.
2014-48 Level 3
Print the puzzles or type the factors on this excel file: 12 Factors 2014-12-01
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.
2014-48 Level 3 Factors

311 and Level 2

/ ivasallay / Leave a comment

  • 311 is a prime number.
  • Prime factorization: 311 is prime.
  • The exponent of prime number 311 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 311 has exactly 2 factors.
  • Factors of 311: 1, 311
  • Factor pairs: 311 = 1 x 311
  • 311 has no square factors that allow its square root to be simplified. √311 ≈ 17.635

How do we know that 311 is a prime number? If 311 were not a prime number, then it would be divisible by at least one prime number less than or equal to √311 ≈ 17.635. Since 311 cannot be divided evenly by 2, 3, 5, 7, 11, 13, or 17, we know that 311 is a prime number.

Here’s a little piece of red and white candy for your enjoyment:

2014-48 Level 2

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

2014-48 Level 2 Factors

310 and Level 1

/ ivasallay / 2 Comments

  • 310 is a composite number.
  • Prime factorization: 310 = 2 x 5 x 31
  • 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 310 has exactly 8 factors.
  • Factors of 310: 1, 2, 5, 10, 31, 62, 155, 310
  • Factor pairs: 310 = 1 x 310, 2 x 155, 5 x 62, or 10 x 31
  • 310 has no square factors that allow its square root to be simplified. √310 ≈ 17.607

This simply wrapped puzzle gift box is very easy to unravel.

2014-48 Level 1

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

2014-48 Level 1 Factors

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

307 and Level 5

/ ivasallay / Leave a comment

  • 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

306 and Level 4

/ ivasallay / 1 Comment

  • 306 is a composite number.
  • Prime factorization: 306 = 2 x 3 x 3 x 17, which can be written 306 = 2 x 3² x 17
  • 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 306 has exactly 12 factors.
  • Factors of 306: 1, 2, 3, 6, 9, 17, 18, 34, 51, 102, 153, 306
  • Factor pairs: 306 = 1 x 306, 2 x 153, 3 x 102, 6 x 51, 9 x 34, or 17 x 18
  • Taking the factor pair with the largest square number factor, we get √306 = (√9)(√34) = 3√34 ≈ 17.493

Happy Thanksgiving!

2014-47 Level 4

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

Here’s a little more about the number 306:

306 = 17 × 18, which means it is the sum of the first 17 even numbers.

  • Thus, 2 + 4 + 6 + 8 + . . .  + 30 + 32 + 34 = 306

It also means that we are halfway between 17² and 18², or halfway between 289 and 324. The average of those two numbers is 306.5.

AND it means that 17² + 18² – 1 = 2(306) = 2(17 × 18)

2014-47 Level 4 Logic