1264 Dum Dums Mystery

Dum Dums have been around since 1924. If you go trick or treating, you will likely get at least one of these popular lollipops. Have fun solving the Dum Dums mystery puzzle I’ve made for you.

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

Here are a few facts about the number 1264:

  • 1264 is a composite number.
  • Prime factorization: 1264 = 2 × 2 × 2 × 2 × 79, which can be written 1264 = 2⁴ × 79
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1264 has exactly 10 factors.
  • Factors of 1264: 1, 2, 4, 8, 16, 79, 158, 316, 632, 1264
  • Factor pairs: 1264 = 1 × 1264, 2 × 632, 4 × 316, 8 × 158, or 16 × 79
  • Taking the factor pair with the largest square number factor, we get √1264 = (√16)(√79) = 4√79 ≈ 35.55278

1264 is the sum of the first twenty-seven prime numbers. That’s all the prime numbers from 2 to 103.

 

 

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1263 Candy Corn Mystery

Candy Corn is a soft traditional Halloween candy. I hope this puzzle is a sweet treat for you to solve! Just write the numbers from one to 10 in the first column and the top row so that the puzzle becomes a sort of multiplication table with the given clues becoming the products of the factors you write.

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

Now I’ll write a little bit about the number 1263:

  • 1263 is a composite number.
  • Prime factorization: 1263 = 3 × 421
  • 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 1263 has exactly 4 factors.
  • Factors of 1263: 1, 3, 421, 1263
  • Factor pairs: 1263 = 1 × 1263 or 3 × 421
  • 1263 has no square factors that allow its square root to be simplified. √1263 ≈ 35.53871

1263 is the hypotenuse of a Pythagorean triple:
87-1260-1263 which is 3 times (29-420-421)

1262 Jack-o-lantern Mystery

I have a week’s worth of Halloween themed mystery level puzzles starting with this jack-o-lantern. Mystery level doesn’t mean it’s difficult, only that I’m not letting you know if it’s tricky or not. You may find this puzzle is a real treat, so give it a try!

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

Here are a few facts about the number 1262:

  • 1262 is a composite number.
  • Prime factorization: 1262 = 2 × 631
  • 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 1262 has exactly 4 factors.
  • Factors of 1262: 1, 2, 631, 1262
  • Factor pairs: 1262 = 1 × 1262 or 2 × 631
  • 1262 has no square factors that allow its square root to be simplified. √1262 ≈ 35.52464

1262 is the sum of the twenty-six prime numbers from 3 to 103.

 

1261 Can You Make a Star out of a Hexagon?

Can you make a star out of a hexagon? If you have 1261 tiny squares arranged as a centered hexagon, you can rearrange those 1261 tiny squares into a six-pointed star as illustrated below!

37 was the last centered hexagonal number that was also a star number.

Here are some more facts about the number 1261:

  • 1261 is a composite number.
  • Prime factorization: 1261 = 13 × 97
  • 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 1261 has exactly 4 factors.
  • Factors of 1261: 1, 13, 97, 1261
  • Factor pairs: 1261 = 1 × 1261 or 13 × 97
  • 1261 has no square factors that allow its square root to be simplified. √1261 ≈ 35.51056

1261 is the sum of two squares two different ways:
30² + 19² = 1261
35² + 6² = 1261

1261 is the hypotenuse of FOUR Pythagorean triples:
420-1189-1261 calculated from 2(35)(6), 35² – 6², 35² + 6²
485-1164-1261 which is (5-12-13) times 97
539-1140-1261 calculated from 30² – 19², 2(30)(19), 30² + 19²
845-936-1261 which is 13 times (65-72-97)

Haunted Forest with 1260 Factor Trees

1260 is the smallest number with 36 factors. That’s a new record. (32 was the old record and was held by both 840 and 1080.)

Often when a number has a lot of factors, we will visit a forest of its factor trees. 1260 certainly deserves such a forest. Since it is just before Halloween, It happens to be a haunted forest. Do you dare to go into such a forest? These three trees are scary enough for me! However, there are MANY more factor trees in that haunted forest! Perhaps if you are brave, you can find some of those factor trees in the haunted forest yourself.

Here’s more about the number 1260:

  • 1260 is a composite number.
  • Prime factorization: 1260 = 2 × 2 × 3 × 3 × 5 × 7, which can be written 1260 = 2² × 3² × 5 × 7.
  • The exponents in the prime factorization are 2, 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) (1 + 1) = 3 × 3 × 2 × 2  = 36. Therefore 1260 has exactly 36 factors.
  • Factors of 1260: 1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 28, 30, 35, 36, 42, 45, 60, 63, 70, 84, 90, 105, 126, 140, 180, 210, 252, 315, 420, 630, 1260
  • Factor pairs: 1260 = 1 × 1260, 2 × 630, 3 × 420, 4 × 315, 5 × 252, 6 × 210, 7 × 180, 9 × 140, 10 × 126, 12 × 105, 14 × 90, 15 × 84, 18 × 70, 20 × 63, 21 × 60, 28 × 45, 30 × 42 or 35 × 36
  • Taking the factor pair with the largest square number factor, we get √1260 = (√36)(√35) = 6√35 ≈ 35.49648

21 × 60 = 1260 The same digits are used on both sides of that equation and that makes 1260 the 19th Friedman number.

1260 is also the sum of the interior angles of a nine-sided polygon. Convex or concave, that is the sum. The concave nonagon below is a good illustration of that fact:

1260 is also the hypotenuse of a Pythagorean triple:
756-1008-1260 which is (3-4-5) times 252

1259 Graveyard Marker

This is the first of a week’s worth of Halloween Find the Factors puzzles. Graveyards are often associated with the holiday. Many graveyards have crosses marking the place where some dearly loved person was laid to rest. This puzzle isn’t very scary. Have fun solving it!

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

Now I’ll share some facts about the number 1259:

  • 1259 is a prime number.
  • Prime factorization: 1259 is prime.
  • The exponent of prime number 1259 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1259 has exactly 2 factors.
  • Factors of 1259: 1, 1259
  • Factor pairs: 1259 = 1 × 1259
  • 1259 has no square factors that allow its square root to be simplified. √1259 ≈ 35.48239

How do we know that 1259 is a prime number? If 1259 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1259 ≈ 35.5. Since 1259 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1259 is a prime number.

1259 is the sum of the twenty-five prime numbers from 5 to 103.

The number after 1259 has thirty-six factors. No wonder 1259 had to settle for 1 and itself being its only factors.

Between prime numbers 1237 and 1277, there are 39 numbers but only two of them are prime numbers. 1259 is one of them. Up to 1277 on the number line, no other segment of the same length has a lower incidence of prime numbers than that!

1250 and Level 6

The clues in one of the columns for this puzzle as well as one of the rows are 9 and 3. You will need to figure out where to put the factors 1, 3, 3, and 9 to make those clues work. You might think it doesn’t matter where you write those factors, but believe me, it does matter. My advice: Don’t start with those clues. Find another logical place to start. Good luck with this one!

Print the puzzles or type the solution in this excel file: 10-factors-1242-1250

Here are some facts about the number 1250:

  • 1250 is a composite number.
  • Prime factorization: 1250 = 2 × 5 × 5 × 5 × 5, which can be written 1250 = 2 × 5⁴
  • The exponents in the prime factorization are 1 and 5. Adding one to each and multiplying we get (1 + 1)(4 + 1) = 2 × 5 = 10. Therefore 1250 has exactly 10 factors.
  • Factors of 1250: 1, 2, 5, 10, 25, 50, 125, 250, 625, 1250
  • Factor pairs: 1250 = 1 × 1250, 2 × 625, 5 × 250, 10 × 125, or 25 × 50
  • Taking the factor pair with the largest square number factor, we get √1250 = (√625)(√2) = 25√2 ≈ 35.35534

1250 is the sum of consecutive squares two different ways:
193 + 197 + 199 + 211 + 223 + 227 = 1250
619 + 631 = 1250

1250 is the sum of two squares THREE different ways:
31² + 17² = 1250
25² + 25² = 1250
35² + 5² = 1250

1250 is the hypotenuse of FOUR Pythagorean triples:
750-1000-1250 which is (3-4-5) times 250,
672-1054-1250 which is 2 times (336-527-625) and
can also be calculated from 31² – 17², 2(31)(17), 31² + 17²,
440-1170-1250 which is 10 times (44-117-125), and
350-1200-1250 which is (7-24-25) times 50 and
can also be calculated from 2(35)(5), 35² – 5², 35² + 5²

1249 and Level 5

Level 5 puzzles can be a little tricky, but if you think about all the factors for the given clues and use logic, you should be able to solve this one. Even if the puzzle tricks you, don’t give up!

Print the puzzles or type the solution in this excel file: 10-factors-1242-1250

Now I’ll share some facts about the number 1249:

  • 1249 is a prime number.
  • Prime factorization: 1249 is prime.
  • The exponent of prime number 1249 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1249 has exactly 2 factors.
  • Factors of 1249: 1, 1249
  • Factor pairs: 1249 = 1 × 1249
  • 1249 has no square factors that allow its square root to be simplified. √1249 ≈ 35.34119

How do we know that 1249 is a prime number? If 1249 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1249 ≈ 35.3. Since 1249 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1249 is a prime number.

1249 is the sum of three consecutive prime numbers:
409 + 419 + 421 = 1249

1249 is the sum of two squares:
32² + 15² = 1249

1249 is the hypotenuse of a Pythagorean triple:
799-960-1249 calculated from 32² – 15², 2(32)(15), 32² + 15²

Here’s another way we know that 1249 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 32² + 15² = 1249 with 32 and 15 having no common prime factors, 1249 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1249 ≈ 35.3. Since 1249 is not divisible by 5, 13, 17, or 29, we know that 1249 is a prime number.

1248 Factor Trees

It is easy to see that 1248 is divisible by 12 and therefore by 2, 3, 4, and 6, but it actually has a lot more factors than that. I decided to celebrate its many factors by making just a few of its many possible factor trees.

Why does 1248 have so many factors? Well, check out its prime factorization:

  • 1248 is a composite number.
  • Prime factorization: 1248 = 2 × 2 × 2 × 2 × 2 × 3 × 13, which can be written 1248 = 2⁵ × 3 × 13
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 × 2 × 2 = 26. Therefore 1248 has exactly 26 factors.
  • Factors of 1248: 1, 2, 3, 4, 6, 8, 12, 13, 16, 24, 26, 32, 39, 48, 52, 78, 96, 104, 156, 208, 312, 416, 624, 1248
  • Factor pairs: 1248 = 1 × 1248, 2 × 624, 3 × 416, 4 × 312, 6 × 208, 8 × 156, 12 × 104, 13 × 96, 16 × 78, 24 × 52, 26 × 48, or 32 × 39
  • Taking the factor pair with the largest square number factor, we get √1248 = (√16)(√78) = 4√78 ≈ 35.32704

1248 is the sum of four consecutive prime numbers:
307 + 311 + 313 + 317 = 1248

1248 is also the hypotenuse of a Pythagorean triple:
480-1152-1248 which is (5-12-13) times 96

1247 Is a Pentagonal Number

Two factors of 1247 make it the 29th pentagonal number. Here’s why:

29(3·29-1)/2 = 29(86)/2 = 29(43) = 1247

Here is an illustration of this pentagonal number featuring a different, but equivalent, formula.  Seeing the pentagonal numbers less than 1247 in the illustration won’t be difficult either.

Here are some more facts about the number 1247:

  • 1247 is a composite number.
  • Prime factorization: 1247 = 29 × 43
  • 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 1247 has exactly 4 factors.
  • Factors of 1247: 1, 29, 43, 1247
  • Factor pairs: 1247 = 1 × 1247 or 29 × 43
  • 1247 has no square factors that allow its square root to be simplified. √1247 ≈ 35.31289

1247 is the sum of consecutive prime numbers two different ways:
It is the sum of the twenty-three prime numbers from 11 to 103.
It is also the sum of seven consecutive primes:
163 + 167 + 173 + 179 + 181 + 191 + 193 = 1247

1247 is the hypotenuse of a Pythagorean triple:
860-903-1247 which is (20-21-29) times 43