1315 Peppermint Stick

/ ivasallay / Leave a comment

 

A red, green, and white peppermint stick can be used to stir hot chocolate or enjoyed as a candy for a long time. Will it take you very long to solve this peppermint stick puzzle? May you find sweet success as you find all the factors!

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

Now I’ll share some facts about the number 1315:

  • 1315 is a composite number.
  • Prime factorization: 1315 = 5 × 263
  • 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 1315 has exactly 4 factors.
  • Factors of 1315: 1, 5, 263, 1315
  • Factor pairs: 1315 = 1 × 1315 or 5 × 263
  • 1315 has no square factors that allow its square root to be simplified. √1315 ≈ 36.26293

1315 is the sum of three consecutive prime numbers:
433 + 439 + 443 = 1315

1315 is the hypotenuse of a Pythagorean triple:
789-1052-1315 which is (3-4-5) times 263

1314 Desmos Art

/ ivasallay / Leave a comment

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:

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

/ ivasallay / Leave a comment

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 in 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

/ ivasallay / Leave a comment

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 in 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

/ ivasallay / Leave a comment

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

As shown in their factor trees below, 1308, 1309, 1310, and 1311 each have three distinct prime numbers in their prime factorizations. They are the smallest set of four consecutive numbers with the same number of prime factors. 1309, 1310, and 1311 are also the smallest three consecutive numbers that have exactly the same number of factors and factor pairs. Thank you OEIS.org for alerting me to those facts.

Finally, although the list of primes is slightly out of order, here’s a fun fact related to 1311 that I saw on Twitter:

 

 

1310 Happy Birthday to My Brother, Andy

/ ivasallay / Leave a comment

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!

Print the puzzles or type the solution in this excel file: 10-factors-1302-1310

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

As shown in their factor trees below, 1308, 1309, 1310, and 1311 each have three distinct prime numbers in their prime factorizations. They are the smallest set of four consecutive numbers with the same number of prime factors. 1309, 1310, and 1311 are also the smallest three consecutive numbers that have exactly the same number of factors and factor pairs. Thank you OEIS.org for alerting me to those facts.

 

1309 Mystery Level

/ ivasallay / Leave a comment

This mystery level puzzle doesn’t look too hard, but is that deceptive? You will have to put your logical mind to work to know for sure.

Print the puzzles or type the solution in this excel file: 10-factors-1302-1310

Now I’d like to share a few facts about 1309:

  • 1309 is a composite number.
  • Prime factorization: 1309 = 7 × 11 × 17
  • 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 1309 has exactly 8 factors.
  • Factors of 1309: 1, 7, 11, 17, 77, 119, 187, 1309
  • Factor pairs: 1309 = 1 × 1309, 7 × 187, 11 × 119, or 17 × 77
  • 1309 has no square factors that allow its square root to be simplified. √1309 ≈ 36.18011

1309 is the hypotenuse of a Pythagorean triple:
616-1155-1309 which is (8-15-17) times 77

As shown in their factor trees below, 1308, 1309, 1310, and 1311 each have three distinct prime numbers in their prime factorizations. They are the smallest set of four consecutive numbers with the same number of prime factors. 1309, 1310, and 1311 are also the smallest three consecutive numbers that have exactly the same number of factors and factor pairs. Thank you OEIS.org for alerting me to those facts.

1308 and Level 6

/ ivasallay / Leave a comment

Can you find the factors that will turn this puzzle into a multiplication table? It’s a level 6 so you might find it to be a challenge. Use logic. Don’t guess and check.

Print the puzzles or type the solution in this excel file: 10-factors-1302-1310

Here are a few facts about the puzzle number, 1308:

  • 1308 is a composite number.
  • Prime factorization: 1308 = 2 × 2 × 3 × 109, which can be written 1308 = 2² × 3 × 109
  • 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 × 2 × 2 = 12. Therefore 1308 has exactly 12 factors.
  • Factors of 1308: 1, 2, 3, 4, 6, 12, 109, 218, 327, 436, 654, 1308
  • Factor pairs: 1308 = 1 × 1308, 2 × 654, 3 × 436, 4 × 327, 6 × 218, or 12 × 109
  • Taking the factor pair with the largest square number factor, we get √1308 = (√4)(√327) = 2√327 ≈ 36.16628

1308 is the hypotenuse of a Pythagorean triple:
720-1092-1308 which is 12 times (60-91-109)

As shown in their factor trees below, 1308, 1309, 1310, and 1311 each have three distinct prime numbers in their prime factorizations. They are the smallest set of four consecutive numbers with the same number of prime factors. Thank you OEIS.org for alerting me to that fact.

 

1307 and Level 5

/ ivasallay / Leave a comment

This puzzle shows 10 of the 100 products in a 10 × 10 multiplication table. Can you figure out where to put the factors? There’s only one way that works!

Print the puzzles or type the solution in this excel file: 10-factors-1302-1310

Here are some facts about the number 1307:

  • 1307 is a prime number.
  • Prime factorization: 1307 is prime.
  • The exponent of prime number 1307 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1307 has exactly 2 factors.
  • Factors of 1307: 1, 1307
  • Factor pairs: 1307 = 1 × 1307
  • 1307 has no square factors that allow its square root to be simplified. √1307 ≈ 36.15245

How do we know that 1307 is a prime number? If 1307 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1307 ≈ 36.2. Since 1307 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1307 is a prime number.

1307 is palindrome 797 in BASE 13 because 7(13²) + 9(13) + 7(1) = 1307

1306 is a Centered Triangular Number

/ ivasallay / Leave a comment

1306 tiny squares can be arranged in a triangle as I have done in the graphic below. This arrangement makes 1306 a centered triangular number:

Here’s more information about the number 1306:

  • 1306 is a composite number.
  • Prime factorization: 1306 = 2 × 653
  • 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 1306 has exactly 4 factors.
  • Factors of 1306: 1, 2, 653, 1306
  • Factor pairs: 1306 = 1 × 1306 or 2 × 653
  • 1306 has no square factors that allow its square root to be simplified. √1306 ≈ 36.13862

OEIS.org informs us that 1306 = 1¹ + 3² + 0³ + 6⁴.

1306 is the sum of two squares:
35² + 9² = 1306

1306 is the hypotenuse of a Pythagorean triple:
630-1144-1306 calculated from 2(35)(9), 35² – 9², 35² + 9²