1263 Candy Corn Mystery

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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

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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?

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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

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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

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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!

1258 Mystery Level

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This mystery level puzzle might start off easy enough, but before too long it will surely be a mystery what your next step should be. Don’t worry, logic can still lead the way on every step, but finding the logic might be trickier than usual.

Print the puzzles or type the solution in this excel file: 12 factors 1251-1258

Here’s some information about the number 1258:

  • 1258 is a composite number.
  • Prime factorization: 1258 = 2 × 17 × 37
  • 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 1258 has exactly 8 factors.
  • Factors of 1258: 1, 2, 17, 34, 37, 74, 629, 1258
  • Factor pairs: 1258 = 1 × 1258, 2 × 629, 17 × 74, or 34 × 37
  • 1258 has no square factors that allow its square root to be simplified. √1258 ≈ 35.4683

1258 is the sum of two squares in two different ways:
27² + 23² = 1258
33² + 13² = 1258

1258 is the hypotenuse of FOUR Pythagorean triples:
200-1242-1258
408-1190-1258
592-1110-1258
858-920-1258

1257 and Level 6

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Both 6 and 12 are allowable common factors of 60 and 12. Likewise, both 8 and 12 are allowable common factors of 96 and 72. In each case, only one of those common factors will work with this puzzle. Don’t guess and check each one. Study the other clues and at least one wrong common factor will be eliminated. Have fun solving it!

Print the puzzles or type the solution in this excel file: 12 factors 1251-1258

Now I’ll write a few things about the number 1257:

  • 1257 is a composite number.
  • Prime factorization: 1257 = 3 × 419
  • 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 1257 has exactly 4 factors.
  • Factors of 1257: 1, 3, 419, 1257
  • Factor pairs: 1257 = 1 × 1257 or 3 × 419
  • 1257 has no square factors that allow its square root to be simplified. √1257 ≈ 35.4542

1257 is the difference of two squares two different ways:
211² – 208² = 1257
629² – 628² = 1257

1257 is palindrome 393 in BASE 19

1256 and Level 5

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Use logic, not guess and check, to find where the numbers from 1 to 12 belong in both the first column and the top row so that the puzzle acts like a multiplication table. Can you do it, or will some of the clues trick you?

Print the puzzles or type the solution in this excel file: 12 factors 1251-1258

Now I’ll share some facts about the number 1256:

  • 1256 is a composite number.
  • Prime factorization: 1256 = 2 × 2 × 2 × 157, which can be written 1256 = 2³ × 157
  • 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 1256 has exactly 8 factors.
  • Factors of 1256: 1, 2, 4, 8, 157, 314, 628, 1256
  • Factor pairs: 1256 = 1 × 1256, 2 × 628, 4 × 314, or 8 × 157
  • Taking the factor pair with the largest square number factor, we get √1256 = (√4)(√314) = 2√314 ≈ 35.44009

1256 is the sum of two squares:
34² + 10² = 1256

1256 is the hypotenuse of a Pythagorean triple:
680-1056-1256 which is 8 times (85-132-157) and
can also be calculated from 2(34)(10), 34² – 10², 34² + 10²

1256 is 888 in BASE 12 because 8(144 + 12 + 1) = 8(157) = 1256

1255 and Level 4

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For this puzzle, you will have to study the twelve clues to figure out where to begin to find your first set of factors. You will then use those factors to figure out the next logical clue to use. Continue the process until all the factors are found. Good luck!

Print the puzzles or type the solution in this excel file: 12 factors 1251-1258

Here is some information about the number 1255:

  • 1255 is a composite number.
  • Prime factorization: 1255 = 5 × 251
  • 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 1255 has exactly 4 factors.
  • Factors of 1255: 1, 5, 251, 1255
  • Factor pairs: 1255 = 1 × 1255 or 5 × 251
  • 1255 has no square factors that allow its square root to be simplified. √1255 ≈ 35.42598

1255 = 251 × 5
Check out those digits on both sides of the equation. Their sameness makes 1255 the18th Friedman number.

1255 is also the hypotenuse of a Pythagorean triple:
753-1004-1255 which is (3-4-5) times 251

1254 and Level 3

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Find the common factor of 8 and 80 so that only numbers from 1 to 12 will be put in the top row of this multiplication table puzzle. Then work down row by row writing the factors of each clue so that the numbers from 1 to 12 appear only once in both the first column and the top row. You can do this!

Print the puzzles or type the solution in this excel file: 12 factors 1251-1258

Here are a few facts about the post number, 1254:

  • 1254 is a composite number.
  • Prime factorization: 1254 = 2 × 3 × 11 × 19
  • 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 1254 has exactly 16 factors.
  • Factors of 1254: 1, 2, 3, 6, 11, 19, 22, 33, 38, 57, 66, 114, 209, 418, 627, 1254
  • Factor pairs: 1254 = 1 × 1254, 2 × 627, 3 × 418, 6 × 209, 11 × 114, 19 × 66, 22 × 57, or 33 × 38
  • 1254 has no square factors that allow its square root to be simplified. √1254 ≈ 35.41186

1254 is the sum of the twenty-four prime numbers from 7 to 103. Do you know what those prime numbers are?