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.

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

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!

 

1298 Another Mystery

Hint #1: Which two clues will use both of the nines?
Hint #2: Which two clues will use both of the sixes?

6, 7, 8, or 12, two of them will use the 1’s. One of those 1’s will be in the first column. They both can’t be. That fact was important when I worked to solve this puzzle.

It won’t be easy, but why don’t you give it a try?

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

It won’t help to solve the puzzle, but I’ll share some information now about our puzzle number, 1298:

  • 1298 is a composite number.
  • Prime factorization: 1298 = 2 × 11 × 59
  • 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 1298 has exactly 8 factors.
  • Factors of 1298: 1, 2, 11, 22, 59, 118, 649, 1298
  • Factor pairs: 1298 = 1 × 1298, 2 × 649, 11 × 118, or 22 × 59
  • 1298 has no square factors that allow its square root to be simplified. √1298 ≈ 36.02777

1298 is the sum of four consecutive prime numbers:
313 + 317 + 331 + 337 = 1298

1297 Mystery

Today’s puzzle has 12 clues, but it still presents quite a mystery. Will you be able to figure out where the factors from 1 to 12 go in the 1st column and the top row of the puzzle, or will you let this mystery stump you?

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

Now I’ll share a little bit of information about the number 1297:

  • 1297 is a prime number.
  • Prime factorization: 1297 is prime.
  • The exponent of prime number 1297 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1297 has exactly 2 factors.
  • Factors of 1297: 1, 1297
  • Factor pairs: 1297 = 1 × 1297
  • 1297 has no square factors that allow its square root to be simplified. √1297 ≈ 36.01389

How do we know that 1297 is a prime number? If 1297 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1297 ≈ 36.01. Since 1297 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1297 is a prime number.

1297 is the sum of two squares:
36² +  1² = 1297

1297 is the hypotenuse of a Pythagorean triple:
72-1295-1297 calculated from 2(36)(1), 36² –  1², 36² +  1²

Here’s another way we know that 1297 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 36² + 1² = 1297 with 36 and 1 having no common prime factors, 1297 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1297 ≈ 36.01. Since 1297 is not divisible by 5, 13, 17, or 29, we know that 1297 is a prime number.

1295 Mystery Level

This pinwheel-shaped puzzle is no little kid’s toy. The logic needed to solve it might be a mystery to see, but it is still there!

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

Here are some facts about the number 1295:

  • 1295 is a composite number.
  • Prime factorization: 1295 = 5 × 7 × 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 1295 has exactly 8 factors.
  • Factors of 1295: 1, 5, 7, 35, 37, 185, 259, 1295
  • Factor pairs: 1295 = 1 × 1295, 5 × 259, 7 × 185, or 35 × 37
  • 1295 has no square factors that allow its square root to be simplified. √1295 ≈ 35.98611

Since 35 × 37 = 1295, we know that 1295 is one number less than 36².

1295 is the hypotenuse of FOUR Pythagorean triples:
399-1232-1295 which is 7 times (57-176-185)
420-1225-1295 which is 35 times (12-35-37)
728-1071-1295 which is 7 times (104-153-185)
777-1036-1295 which is (3-4-5) times 259

1288 Mystery Puzzle

How difficult is this mystery level puzzle? That is part of the mystery, but I assure you that if you use logic and basic multiplication facts you can find the unique solution.

Print the puzzles or type the solution in this excel file: 10-factors-1281-1288

That was puzzle #1288. Now I’ll tell you some facts about the number 1288.

  • 1288 is a composite number.
  • Prime factorization: 1288 = 2 × 2 × 2 × 7 × 23, which can be written 1288 = 2³ × 7 × 23
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1288 has exactly 16 factors.
  • Factors of 1288: 1, 2, 4, 7, 8, 14, 23, 28, 46, 56, 92, 161, 184, 322, 644, 1288
  • Factor pairs: 1288 = 1 × 1288, 2 × 644, 4 × 322, 7 × 184, 8 × 161, 14 × 92, 23 × 56, or 28 × 46
  • Taking the factor pair with the largest square number factor, we get √1288 = (√4)(√322) = 2√322 ≈ 37.88872

1288 has four factor pairs that contain only even factors so 1288 can be written as the difference of two squares four different ways:
323² – 321² = 1288
163² – 159² = 1288
53² – 39² = 1288
37² – 9² = 1288