Prime factorization

1313 Virgács and St. Nickolas Day

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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

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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

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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

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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

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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

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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.

 

1306 is a Centered Triangular Number

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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

1305 and Level 4

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This level 4 puzzle has clues taken from a standard multiplication table, but the factors of those clues are not in their usual places. Can you figure out where the numbers 1 to 10 belong in this multiplication table?

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

Now I’ll share some information about the puzzle’s number, 1305:

  • 1305 is a composite number.
  • Prime factorization: 1305 = 3 × 3 × 5 × 29, which can be written 2905 = 3² × 5 × 29
  • 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 1305 has exactly 12 factors.
  • Factors of 1305: 1, 3, 5, 9, 15, 29, 45, 87, 145, 261, 435, 1305
  • Factor pairs: 1305 = 1 × 1305, 3 × 435, 5 × 261, 9 × 145, 15 × 87, or 29 × 45
  • Taking the factor pair with the largest square number factor, we get √1305 = (√9)(√145) = 3√145 ≈ 36.12478.

1305 is the sum of two squares in two ways:
36² + 3² = 1305
27² + 24² = 1305

153-1296-1305 calculated by 27² – 24², 2(27)(24), 27² + 24²
216-1287-1305 calculated by 2(36)(3), 36² – 3², 36² + 3²
783-1044-1305 which is (3-4-5) times 261
900-945-1305 which is (20-21-29) times 45

1304 and Level 3

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Which factor pairs of 45 and 18 have only numbers from 1 to 10 in them? Answer that question, put the factors in the appropriate places, and then work your way down this level 3 puzzle cell by cell until you’ve solved it.

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

Here are a few facts about the number 1304:

  • 1304 is a composite number.
  • Prime factorization: 1304 = 2 × 2 × 2 × 163, which can be written 1304 = 2³ × 163
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1304 has exactly 8 factors.
  • Factors of 1304: 1, 2, 4, 8, 163, 326, 652, 1304
  • Factor pairs: 1304 = 1 × 1304, 2 × 652, 4 × 326, or 8 × 163
  • Taking the factor pair with the largest square number factor, we get √1304 = (√4)(√326) = 2√326 ≈ 36.11094

1304 is the difference of two squares two ways:
327² – 325² = 1304
165² – 161² = 1304

1301 Cool Patterns of Centered Square Numbers

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1301 is a centered square number because it is the sum of two consecutive square numbers. It also happens to be exactly 100 more than the previous centered square number.

1301 is special because it is the sum of two consecutive squares, but it is not the only sum of two squares that is 100 more than another sum of two squares. Here is a chart of some of those squares with some other information. What patterns do you see?

What role did the triangular numbers play on that chart? The pattern you see of 4, 12, 24, etc. is not new. Look at this chart of centered square numbers below. Below each centered square number (in blue) are related sums of squares. The difference of those related sums and the centered square above it follows that same 4, 12, 24, . . .  pattern.

When you look just at centered square numbers, there is a different pattern for them as well. For example, if you know that 1301 is the 26th centered square number, then you can be sure that 1301 + 4(26) = 1405 will be the next centered square number.

When the chart above is “bent in half” so the list of centered square numbers begins at the lower left and ends at the lower right, there is yet another pattern. Look at the differences horizontally to see a cool pattern:

I’ve made it clear that centered square numbers are made when consecutive square numbers are added, such as 26² + 25² = 1301, but what does one look like? Here’s a picture of 1301 tiny little squares arranged as a centered square.

There are 100 tiny green squares around it because it is 100 more than the previous centered square number.

Here are some more facts about the number 1301:

  • 1301 is a prime number and the first prime in a prime triplet.
  • Prime factorization: 1301 is prime.
  • The exponent of prime number 1301 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1301 has exactly 2 factors.
  • Factors of 1301: 1, 1301
  • Factor pairs: 1301 = 1 × 1301
  • 1301 has no square factors that allow its square root to be simplified. √1301 ≈ 36.06938

How do we know that 1301 is a prime number? If 1301 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1301 ≈ 36.1. Since 1301 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1301 is a prime number.

Every centered square number is a hypotenuse of a Pythagorean triple. For example,
primitive 51-1300-1301 is calculated from 26² – 25², 2(26)(25), 26² + 25²

Here’s another way we know that 1301 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 26² + 25² = 1301 with 26 and 25 having no common prime factors, 1301 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1301 ≈ 36.1. Since 1301 is not divisible by 5, 13, 17, or 29, we know that 1301 is a prime number.