1366 Fractions Acting Improperly

In elementary school, we learned about improper fractions. Should we call them that? Is it even possible for any kind of number to be IMPROPER? They are simply fractions greater than one. I’ve recently heard the term fraction form used, and ever since I’ve made a point of saying that fractions greater than one are in fraction form.

On Twitter, I’ve found a few people who also don’t like using the word improper to describe any fraction.

This first tweet has a link explaining why it is improper to use the term improper fraction:

I hope that you will consider not labeling any fraction as improper, as well!

Now I’ll write a little bit about the number 1366:

  • 1366 is a composite number.
  • Prime factorization: 1366 = 2 × 683
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1366 has exactly 4 factors.
  • Factors of 1366: 1, 2, 683, 1366
  • Factor pairs: 1366 = 1 × 1366 or 2 × 683
  • 1366 has no square factors that allow its square root to be simplified. √1366 ≈ 36.95944

1366 is also the sum of the twenty-six prime number from 5 to 107. Do you know what all those prime numbers are?

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1365 Shamrock Mystery

Beautiful shamrocks with their three heart-shaped leaves are not difficult to find. Finding the factors in this shamrock-shaped puzzle might be a different story.  Sure, it might start off to be easy, but after a while, you might find it a wee bit more difficult, unless, of course, the luck of the Irish is with you!

Now I’ll share some information about the number 1365:

  • 1365 is a composite number.
  • Prime factorization: 1365 = 3 × 5 × 7 × 13
  • 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 × 2 × 2 × 2 = 16. Therefore 1365 has exactly 16 factors.
  • Factors of 1365: 1, 3, 5, 7, 13, 15, 21, 35, 39, 65, 91, 105, 195, 273, 455, 1365
  • Factor pairs: 1365 = 1 × 1365, 3 × 455, 5 × 273, 7 × 195, 13 × 105, 15 × 91, 21 × 65, or 35 × 39
  • 1365 has no square factors that allow its square root to be simplified. √1365 ≈ 36.94591

1365 is the hypotenuse of FOUR Pythagorean triples:
336-1323-1365 which is 21 times (16-63-65)
525-1260-1365 which is (5-12-13) times 105
693-1176-1365 which is 21 times (33-56-65)
819-1092-1365 which is (3-4-5) times 273

1365 looks interesting in some other bases:
It’s 10101010101 in BASE 2,
111111 in BASE 4,
2525 in BASE 8, and
555 in BASE 16

I’m feeling pretty lucky that I noticed all those fabulous number facts! If you haven’t been so lucky finding the factors of the puzzle, the same puzzle but with more clues might help:

1350 Logic is at the Heart of This Puzzle

By simply changing two clues of that recently published puzzle that I rejected, I was able to create a love-ly puzzle that can be solved entirely by logic. Can you figure out where to put the numbers from 1 to 12 in each of the four outlined areas that divide the puzzle into four equal sections? If you can, my heart might just skip a beat!

Now I’ll tell you a few things about the number 1350:

  • 1350 is a composite number.
  • Prime factorization: 1350 = 2 × 3 × 3 × 3 × 5 × 5, which can be written 1350 = 2 × 3³ × 5²
  • The exponents in the prime factorization are 1, 3 and 2. Adding one to each and multiplying we get (1 + 1)(3 + 1)(2 + 1) = 2 × 4 × 3 = 24. Therefore 1350 has exactly 24 factors.
  • Factors of 1350: 1, 2, 3, 5, 6, 9, 10, 15, 18, 25, 27, 30, 45, 50, 54, 75, 90, 135, 150, 225, 270, 450, 675, 1350
  • Factor pairs: 1350 = 1 × 1350, 2 × 675, 3 × 450, 5 × 270, 6 × 225, 9 × 150, 10 × 135, 15 × 90, 18 × 75, 25 × 54, 27 × 50 or 30 × 45
  • Taking the factor pair with the largest square number factor, we get √1350 = (√225)(√6) = 15√6 ≈ 36.74235

1350 is the sum of consecutive prime numbers two ways:
It is the sum of the fourteen prime numbers from 67 to 131, and
673 + 677 = 1350

1350 is the hypotenuse of two Pythagorean triples:
810-1080-1350 which is (3-4-5) times 270
378-1296-1350 which is (7-24-25) times 54

1350 is also the 20th nonagonal number because 20(7 · 20 – 5)/2 = 1350

1314 Desmos Art

A teacher at my school had his students graph some polynomials and their inverses. I got to help some of his students with their graphs. After seeing the beautiful symmetry of the graphs together, I excitedly exclaimed to a few of the students, “Isn’t this a cool assignment?”

During my lunch, I put one of the graphs, its inverse, and some of their translations on Desmos and made a simple but lovely piece of art in the process. 

Before I was done, I showed it to a couple of students. One of them asked, “Are you saying that math can create art?” I loved replying, “Yes, it can!” Now that student wants to create some works of art, too. It was a privilege to show her how to use Desmos.

These are the inequalities I used to make my work of art:

MANY teachers have figured out that students could learn a lot about functions and their graphs by using Desmos to create drawings, pictures, or artwork. For example, look at this tweet and link shared by Chris Bolognese:

Here’s the final @desmos coloring book from my amazing Precalculus students. We are headed to the lower school next week to join them in some coloring fun. Pictures to come! https://t.co/iejf2RYVWX @ColumbusAcademy @CAVikesSTEAM @mathequalslove #iteachmath— Chris Bolognese (@EulersNephew) December 8, 2018

Now I’ll share some facts about the number 1314:

  • 1314is a composite number.
  • Primefactorization: 1314= 2 × 3 × 3 × 73,which can be written 1314 = 2 × 3² × 73
  • The exponents inthe prime factorization are 1, 2, and 1. Adding one to each and multiplying weget (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1314has exactly 12 factors.
  • Factors of 1314:1, 2, 3, 6, 9, 18, 73, 146, 219, 438, 657, 1314
  • Factor pairs: 1314= 1 × 1314,2 × 657, 3 × 438, 6 × 219, 9 × 146, or 18 × 73 
  • Taking the factorpair with the largest square number factor, we get √1314= (√9)(√146) = 3√146 ≈ 36.24914

1314 is the sum of two squares:
33² + 15² = 1314

1314 is the hypotenuse of a Pythagorean triple:
864-990-1314 which is 18 times (48-55-73) and
can also be calculated from 33² – 15², 2(33)(15), 33² + 15²

1313 Virgács and St. Nickolas Day

6 December is Saint Nickolas Day. Children in Hungary and other places in Europe wake up to find candy and virgács in their boots. You can read more about this wonderful tradition in Jön a Mikulás (Santa is Coming) or Die Feier des Weihnachtsmanns (The Celebration of Santa Claus). Today’s puzzle represents the virgács given to children who have been even the least bit naughty during the current year.

Print the puzzles or type the solution in this excel file: 12 factors 1311-1319

Now I’ll write a little bit about the number 1313:

  • 1313 is a composite number.
  • Prime factorization: 1313 = 13 × 101
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1313 has exactly 4 factors.
  • Factors of 1313: 1, 13, 101, 1313
  • Factor pairs: 1313 = 1 × 1313 or 13 × 101
  • 1313 has no square factors that allow its square root to be simplified. √1313 ≈ 36.23534

1313 is the sum of consecutive prime numbers three different ways:
It is the sum of the twenty-one prime numbers from 19 to 107.
It is the sum of eleven consecutive primes:
97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 = 1313,
and it is the sum of seven consecutive prime numbers:
173 + 179 + 181 + 191 + 193 + 197 + 199 = 1313

1313 is the sum of two squares two different ways:
32² + 17² = 1313
28² +  23² = 1313

1313 is the hypotenuse of FOUR Pythagorean triples:
255-1288-1313 calculated from 28² –  23², 2(28)(23), 28² +  23²
260-1287-1313 which is 13 times (20-99-101)
505-1212-1313 which is (5-12-13) times 101
735-1088-1313 calculated from 32² – 17², 2(32)(17), 32² + 17²

1312 Fill This Boot with Candy

On the 5th of December, many children in the world prepare for a visit from Saint Nickolas by polishing their boots. Hopefully, they have been good boys or girls all year and will find those boots filled the next morning with their favorite candies. Here’s a boot-shaped puzzle for you to solve.

Print the puzzles or type the solution in this excel file: 12 factors 1311-1319

Now I’ll share some information about the number 1312:

  • 1312 is a composite number.
  • Primefactorization: 1312 = 2 × 2 × 2 × 2 × 2 × 41, which can be written 1312 = 2⁵ × 41
  • The exponents inthe prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 1312 has exactly 12 factors.
  • Factors of 1312: 1, 2, 4, 8, 16, 32, 41, 82, 164, 328, 656, 1312
  • Factor pairs: 1312 = 1 × 1312, 2 × 656, 4 × 328, 8 × 164, 16 × 82, or 32 × 41
  • Taking the factor pair with the largest square number factor, we get √1312 = (√16)(√82) = 4√82 ≈ 36.22154

1312 is the sum of consecutive prime numbers two different ways:
It is the sum of the sixteen prime numbers from 47 to 113. Also,
prime numbers 653 + 659 = 1312

1312 is the sum of two squares:
36² + 4² = 1312

1312 is also the hypotenuse of a Pythagorean triple:
288-1280-1312 which is 32 times (9-40-41)

1311 Little Square Candies

Here’s a puzzle made with some sweet squares. The nine clues in it are all you need to find the factors and complete the entire “mixed-up” multiplication table. 

Print the puzzles or type the solution in this excel file: 12 factors 1311-1319

Now I’ll write some facts about the number 1311:

  • 1311 is a composite number.
  • Prime factorization: 1311 = 3 × 19 × 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 × 2 × 2 = 8. Therefore 1311 has exactly 8 factors.
  • Factors of 1311: 1, 3, 19, 23, 57, 69, 437, 1311
  • Factor pairs: 1311 = 1 × 1311, 3 × 437, 19 × 69, or 23 × 57
  • 1311 has no square factors that allow its square root to be simplified. √1311≈ 36.20773

1311 is the sum of five consecutive prime numbers:
251 + 257 + 263 + 269 + 271 = 1311

1310 Happy Birthday to My Brother, Andy

 

Today is my brother’s birthday. He likes puzzles so I’ve made him a tough, challenging one. Still, he’ll probably figure it out in no time. Happy birthday, Andy!

Like always, I’ll write what I’ve learned about a number. This time it’s 1310’s turn.

  • 1310 is a composite number.
  • Prime factorization: 1310 = 2 × 5 × 131
  • 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 × 2 × 2 = 8. Therefore 1310 has exactly 8 factors.
  • Factors of 1310: 1, 2, 5, 10, 131, 262, 655, 1310
  • Factor pairs: 1310 = 1 × 1310, 2 × 655, 5 × 262, or 10 × 131
  • 1310 has no square factors that allow its square root to be simplified. √1310 ≈ 36.19392

1310 is the hypotenuse of a Pythagorean triple:
786-1048-1310 which is (3-4-5) times 262

 

1299 Is This Puzzle a Real Turkey?

Happy Thanksgiving, everyone!

Turkeys run but they cannot hide. They all will eventually end up on somebodies’ table. There doesn’t seem to be much of a mystery about that, but I’ve created a mystery level puzzle for today anyway. I promise it can be solved using logic and the basic facts in a 12 × 12 multiplication table.

Now I’ll share some facts about the number 1299:

  • 1299 is a composite number.
  • Prime factorization: 1299 = 3 × 433
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 1299 has exactly 4 factors.
  • Factors of 1299: 1, 3, 433, 1299
  • Factor pairs: 1299 = 1 × 1299 or 3 × 433
  • 1299 has no square factors that allow its square root to be simplified. √1299 ≈ 36.04164

1299 is the hypotenuse of a Pythagorean triple:
435-1224-1299 which is 3 times (145-408-433)

Stetson.edu informs us that 8¹²⁹⁹ ≈ 1299 × 10¹¹⁷⁰. You can see it for yourself on a computer calculator!

 

1270 What’s Brewing on My 5-Year Blogiversary

As Halloween approaches, I remember that five years ago today, I hit the publish button for the first time, and my puzzles became available for anyone with an internet connection to use.

Today’s puzzle looks a little bit like a cauldron. What’s brewing on my 5-year blogiversary?

Print the puzzles or type the solution in this excel file: 10-factors-1259-1270

I continue to be very grateful to WordPress and the WordPress community for making blogging and publishing easy and enjoyable. I am also very grateful to my readers who have done so much to make this blog grow.

I’m a lot busier now than I was five years ago. Besides blogging, I have a full-time job and a part-time job. I like both of these jobs because I like helping students understand mathematics better. Sometimes I don’t have the time I would like to work on my blog. Nevertheless, I still have blogging goals I want to reach so lately I find myself playing catch-up more often than not.

Now I’ll write a little about the number 1270:

  • 1270 is a composite number.
  • Prime factorization: 1270 = 2 × 5 × 127
  • 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 × 2 × 2 = 8. Therefore 1270 has exactly 8 factors.
  • Factors of 1270: 1, 2, 5, 10, 127, 254, 635, 1270
  • Factor pairs: 1270 = 1 × 1270, 2 × 635, 5 × 254, or 10 × 127
  • 1270 has no square factors that allow its square root to be simplified. √1270 ≈ 35.63706

1270 is the hypotenuse of a Pythagorean triple:
762-1016-1270 which is (3-4-5) times 254