1365 Shamrock Mystery

Beautiful shamrocks with their three heart-shaped leaves are not difficult to find. Finding the factors in this shamrock-shaped puzzle might be a different story.  Sure, it might start off to be easy, but after a while, you might find it a wee bit more difficult, unless, of course, the luck of the Irish is with you!

Now I’ll share some information about the number 1365:

  • 1365 is a composite number.
  • Prime factorization: 1365 = 3 × 5 × 7 × 13
  • 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 1365 has exactly 16 factors.
  • Factors of 1365: 1, 3, 5, 7, 13, 15, 21, 35, 39, 65, 91, 105, 195, 273, 455, 1365
  • Factor pairs: 1365 = 1 × 1365, 3 × 455, 5 × 273, 7 × 195, 13 × 105, 15 × 91, 21 × 65, or 35 × 39
  • 1365 has no square factors that allow its square root to be simplified. √1365 ≈ 36.94591

1365 is the hypotenuse of FOUR Pythagorean triples:
336-1323-1365 which is 21 times (16-63-65)
525-1260-1365 which is (5-12-13) times 105
693-1176-1365 which is 21 times (33-56-65)
819-1092-1365 which is (3-4-5) times 273

1365 looks interesting in some other bases:
It’s 10101010101 in BASE 2,
111111 in BASE 4,
2525 in BASE 8, and
555 in BASE 16

I’m feeling pretty lucky that I noticed all those fabulous number facts! If you haven’t been so lucky finding the factors of the puzzle, the same puzzle but with more clues might help:

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1299 Is This Puzzle a Real Turkey?

Happy Thanksgiving, everyone!

Turkeys run but they cannot hide. They all will eventually end up on somebodies’ table. There doesn’t seem to be much of a mystery about that, but I’ve created a mystery level puzzle for today anyway. I promise it can be solved using logic and the basic facts in a 12 × 12 multiplication table.

Now I’ll share some facts about the number 1299:

  • 1299 is a composite number.
  • Prime factorization: 1299 = 3 × 433
  • 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 1299 has exactly 4 factors.
  • Factors of 1299: 1, 3, 433, 1299
  • Factor pairs: 1299 = 1 × 1299 or 3 × 433
  • 1299 has no square factors that allow its square root to be simplified. √1299 ≈ 36.04164

1299 is the hypotenuse of a Pythagorean triple:
435-1224-1299 which is 3 times (145-408-433)

Stetson.edu informs us that 8¹²⁹⁹ ≈ 1299 × 10¹¹⁷⁰. You can see it for yourself on a computer calculator!

 

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

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 1268:

  • 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)

 

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