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

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1269 Tangram Witch Puzzle

If you have the seven tangram pieces, then you can create this Halloween witch riding across the moon (or a paper plate). Best Witches creating all kinds of things with those fabulous tiles!

In case you would like to know some facts about the number 1269, here’s what I’ve learned:

  • 1269 is a composite number.
  • Prime factorization: 1269 = 3 × 3 × 3 × 47, which can be written 1269 = 3³ × 47
  • 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 1269 has exactly 8 factors.
  • Factors of 1269: 1, 3, 9, 27, 47, 141, 423, 1269
  • Factor pairs: 1269 = 1 × 1269, 3 × 423, 9 × 141, or 27 × 47
  • Taking the factor pair with the largest square number factor, we get √1269 = (√9)(√141) = 3√141 ≈ 35.62303

1269 is the difference of two squares four different ways:
37² – 10² = 1269
75² – 66² = 1269
213² – 210² = 1269
635² – 634² = 1269

1268 Halloween Cat Mystery

Cats can be quite mysterious. They are a favorite pet for many every day, even though suspicious stories abound about them on Halloween. Can you solve the mystery of this cat-like puzzle?

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

Now I’ll share a few facts about the number 317:

  • 1268 is a composite number.
  • Prime factorization: 1268 = 2 × 2 × 317, which can be written 1268 = 2² × 317
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 1268 has exactly 6 factors.
  • Factors of 1268: 1, 2, 4, 317, 634, 1268
  • Factor pairs: 1268 = 1 × 1268, 2 × 634, or 4 × 317
  • Taking the factor pair with the largest square number factor, we get √1268 = (√4)(√317) = 2√317 ≈ 35.60899

28² + 22² = 1268

1268 is the hypotenuse of a Pythagorean triple:
300-1232-1268 calculated from 28² – 22², 2(28)(22), 28² + 22².
It is also 4 times (75-308-317)

1267 Frankenstein Mystery

There are legends of Dr. Frankenstein creating a monster years ago. Nowadays Frankenstein’s Monster can often be seen walking through neighborhoods on Halloween night. This puzzle looks a little bit like him.

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

But if you take all the color away, he looks completely different and quite harmless:

Now I’ll share some information about the number 1267:

  • 1267 is a composite number.
  • Prime factorization: 1267 = 7 × 181
  • 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 1267 has exactly 4 factors.
  • Factors of 1267: 1, 7, 181, 1267
  • Factor pairs: 1267 = 1 × 1267 or 7 × 181
  • 1267 has no square factors that allow its square root to be simplified. √1267 ≈ 35.59494

1267 is the sum of nine consecutive prime numbers:
113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 + 163 = 1267

1267 is the hypotenuse of a Pythagorean triple:
133-1260-1267 which is 7 times (19-180-181)

 

1266 is a Centered Pentagonal Number

1266 is the 23rd centered pentagonal number because 5(22)(23)/2 + 1 = 1266. The graphic below shows 1266 tiny dots arranged into a pentagonal shape and that 1266 is one more than five times the 22nd triangular number.

Here are some more facts about the number 1266:

  • 1266 is a composite number.
  • Prime factorization: 1266 = 2 × 3 × 211
  • 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 1266 has exactly 8 factors.
  • Factors of 1266: 1, 2, 3, 6, 211, 422, 633, 1266
  • Factor pairs: 1266 = 1 × 1266, 2 × 633, 3 × 422, or 6 × 211
  • 1266 has no square factors that allow its square root to be simplified. √1266 ≈ 35.58089

1266 is also the sum of ten consecutive prime numbers:
103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 = 1266

 

1265 More Candy Corn

People don’t each candy corn every day and the only corn in it is corn syrup.  We usually only see it this time of year. Here’s a puzzle with some candy corn for you to enjoy:

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

  • 1265 is a composite number.
  • Prime factorization: 1265 = 5 × 11 × 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 1265 has exactly 8 factors.
  • Factors of 1265: 1, 5, 11, 23, 55, 115, 253, 1265
  • Factor pairs: 1265 = 1 × 1265, 5 × 253, 11 × 115, or 23 × 55
  • 1265 has no square factors that allow its square root to be simplified. √1265 ≈ 35.56684

1265 is the sum of the fifteen prime numbers from 53 to 113.

1265 is the hypotenuse of a Pythagorean triple:
759-1012-1265 which is (3-4-5) times 253

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.

 

 

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)