1303 and Level 2

Multiplication tables usually have facts up to 10 × 10 = 100 or possibly 12 × 12 = 144. Numbers like 64 and 25 appear only once in those multiplication tables. Those two clues can help you get a good start solving this level 2 puzzle.

Print the puzzles or type the solution in this excel file: 10-factors-1302-1310

Now I’ll share a few facts about the number 1303:

  • 1303 is a prime number.
  • Prime factorization: 1303 is prime.
  • The exponent of prime number 1303 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1303 has exactly 2 factors.
  • Factors of 1303: 1, 1303
  • Factor pairs: 1303 = 1 × 1303
  • 1303 has no square factors that allow its square root to be simplified. √1303 ≈ 36.09709

How do we know that 1303 is a prime number? If 1303 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1303 ≈ 36.1. Since 1303 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1303 is a prime number.

1303 is the sum of three consecutive primes:
431 + 433 + 439 = 1303

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1291 and Level 3

If you think of the common factors of 25 and 55, then you have actually started to solve this puzzle! Start at the top of the puzzle and work your way down. Try it!

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

Here are some facts about the number 1291

  • 1291 is a prime number.
  • Prime factorization: 1291 is prime.
  • The exponent of prime number 1291 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1291 has exactly 2 factors.
  • Factors of 1291: 1, 1291
  • Factor pairs: 1291 = 1 × 1291
  • 1291 has no square factors that allow its square root to be simplified. √1291 ≈ 35.93049

How do we know that 1291 is a prime number? If 1291 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1291 ≈ 35.9. Since 1291 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1291 is a prime number.

1291 is also palindrome 1D1 in BASE 30 because 30² + 13(30) + 1 = 1291

 

1289 and Level 1

You might think this is a very easy puzzle, but for some people, it will be challenging, and will hopefully help them learn some multiplication facts better.

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

Since this is puzzle number 1289, I’ll share some facts about that number:

  • 1289 is a prime number.
  • Prime factorization: 1289 is prime.
  • The exponent of prime number 1289 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1289 has exactly 2 factors.
  • Factors of 1289: 1, 1289
  • Factor pairs: 1289 = 1 × 1289
  • 1289 has no square factors that allow its square root to be simplified. √1289 ≈ 35.90265

How do we know that 1289 is a prime number? If 1289 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1289 ≈ 35.9. Since 1289 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1289 is a prime number.

1289 is the sum of two squares:
35² + 8² = 1289

1289 is the hypotenuse of a Pythagorean triple:
560-1161-1289 calculated from 2(35)(8), 35² – 8², 35² + 8²

Here’s another way we know that 1289 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 8² = 1289 with 35 and 8 having no common prime factors, 1289 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1289 ≈ 35.9. Since 1289 is not divisible by 5, 13, 17, or 29, we know that 1289 is a prime number.

1279 and Level 5

Can you find the factors for the one and only solution to this puzzle that will make the clues be the products of those factors? Sure you can! Give it a try, and don’t give up no matter what!

Print the puzzles or type the solution in this excel file: 12 factors 1271-1280

Now I’ll write a little about the number 1279:

  • 1279 is a prime number.
  • Prime factorization: 1279 is prime.
  • The exponent of prime number 1279 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1279 has exactly 2 factors.
  • Factors of 1279: 1, 1279
  • Factor pairs: 1279 = 1 × 1279
  • 1279 has no square factors that allow its square root to be simplified. √1279 ≈ 35.76311

How do we know that 1279 is a prime number? If 1279 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1279 ≈ 35.7. Since 1279 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1279 is a prime number.

1279 is a prime number that helps us find another VERY big prime number. As Stetson.edu informs us, 2¹²⁷⁹ – 1 is also a prime number and is known as a Mersenne Prime.

 

1277 Strată Bolyai János in Timișoara, Romania

Around the turn of the 20th century, Bolyai Farkás taught mathematics at a university in Transylvania.  One day he was too sick to teach, so he sent his mathematically gifted 13-year-old son, János, to teach his classes! As you might imagine, János became quite the mathematician in his own right.

Ninety-five years ago today Bolyai János went to Timișoara, Romania to announce his findings concerning geometry’s fifth postulate. For centuries it was argued that this parallel lines postulate could probably be proved using the previous four of Euclid’s postulates, and it should, therefore, be considered a theorem rather than a postulate. Bolyai János proved that it is indeed something that must be assumed rather than proven, because, by assuming it wasn’t necessary, he was able to create a new and very much non-Euclidean geometry, now known as hyperbolic geometry or Bolyai–Lobachevskian geometry.

Last summer I was walking with some family members through a shopping area behind the opera house in Timișoara, Romania. Suddenly my son, David, excitedly shouted, “Mom, look!” There we stood in front of a street sign marking the strată named for Bolyai János! Here is a picture of me in front of that street sign.

Under his image are several plaques. The first is a replica of part of his proof. Underneath are plaques with a quote from him translated into several languages. Perhaps your favorite language is among them. Here is a close-up of the plaques:

The plaque at the bottom is in English, “From nothing I have created a new and another world. It was with these words that on November 3, 1823, Janos Bolyai announced from Timișoara the discovery of the fundamental formula of the first non-Euclidean geometry.”

We did not get to visit the university named for Bolyai János, but I am thrilled that my son spotted this historic location!

Now I’ll write a little about the number 1277:

  • 1277 is a prime number.
  • Prime factorization: 1277 is prime.
  • The exponent of prime number 1277 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1277 has exactly 2 factors.
  • Factors of 1277: 1, 1277
  • Factor pairs: 1277 = 1 × 1277
  • 1277 has no square factors that allow its square root to be simplified. √1277 ≈ 35.73514

How do we know that 1277 is a prime number? If 1277 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1277 ≈ 35.7. Since 1277 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1277 is a prime number.

1277 is the sum of two squares:
34² + 11² = 1277

1277 is the hypotenuse of a Pythagorean triple:
748-1035-1277 calculated from 2(34)(11), 34² – 11², 34² + 11²

Here’s another way we know that 1277 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 34² + 11² = 1277 with 34 and 11 having no common prime factors, 1277 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1277 ≈ 35.7. Since 1277 is not divisible by 5, 13, 17, or 29, we know that 1277 is a prime number.

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.

1237 and Level 5

Level 5 puzzles always have at least one set of clues with more than one possible common factor. Still only one of those factors will actually work with all the rest of the clues. Can you use logic to find the factors needed to solve this puzzle?

Print the puzzles or type the solution in this excel file: 12 factors 1232-1241

Let’s look at some facts about the number 1237:

  • 1237 is a prime number.
  • Prime factorization: 1237 is prime.
  • The exponent of prime number 1237 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1237 has exactly 2 factors.
  • Factors of 1237: 1, 1237
  • Factor pairs: 1237 = 1 × 1237
  • 1237 has no square factors that allow its square root to be simplified. √1237 ≈ 35.17101

How do we know that 1237 is a prime number? If 1237 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1237 ≈ 35.1. Since 1237 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1237 is a prime number.

1237 is the sum of two squares:
34² + 9² = 1237

1237 is the hypotenuse of a Pythagorean triple:
612-1075-1237 calculated from 2(34)(9), 34² – 9², 34² + 9²

Here’s another way we know that 1237 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 34² + 9² = 1237 with 34 and 9 having no common prime factors, 1237 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1237 ≈ 35.1. Since 1237 is not divisible by 5, 13, 17, or 29, we know that 1237 is a prime number.

1231 Mystery Level Puzzle

For almost all the sets of clues in this puzzle, there is more than one permissible common factor. That makes the puzzle a little tricky, but with care, you can still solve it using logic and your knowledge of the basic 10×10 multiplication table. Good luck!

Print the puzzles or type the solution in this excel file: 10-factors-1221-1231

Now I’ll tell you a little bit about the number 1231:

  • 1231 is a prime number.
  • Prime factorization: 1231 is prime.
  • The exponent of prime number 1231 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1231 has exactly 2 factors.
  • Factors of 1231: 1, 1231
  • Factor pairs: 1231 = 1 × 1231
  • 1231 has no square factors that allow its square root to be simplified. √1231 ≈ 35.08561

How do we know that 1231 is a prime number? If 1231 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1231 ≈ 35.1. Since 1231 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1231 is a prime number.

1231 is a palindrome in a couple of bases:
It’s A1A in BASE 11 because 10(11²) + 1(11) + 10(1) = 1231, and
it’s 1B1 in BASE 30 because 1(30²) + 11(30) + 1(1) = 1231

1229 and Level 6

The only common factors permitted for 32 and 40 in this puzzle are 4 and 8, but which one will work for this puzzle? Likewise, you must decide if 3 or 6 is the right common factor for 18 and 30. Don’t guess which factor to use. Study the other clues and let logic guide your decisions until the unique solution is found. Have fun with this one!

Print the puzzles or type the solution in this excel file: 10-factors-1221-1231

This is my 1229th post, so I will tell you a little bit about the number 1229:

  • 1229 is a prime number.
  • Prime factorization: 1229 is prime.
  • The exponent of prime number 1229 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1229 has exactly 2 factors.
  • Factors of 1229: 1, 1229
  • Factor pairs: 1229 = 1 × 1229
  • 1229 has no square factors that allow its square root to be simplified. √1229 ≈ 35.057

How do we know that 1229 is a prime number? If 1229 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1229 ≈ 35.1. Since 1229 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1229 is a prime number.

1229 is the sum of three consecutive prime numbers:
401 + 409 + 419 = 1229

1229 is the sum of two square numbers:
35² + 2²  = 1229

1229 is the hypotenuse of a primitive Pythagorean triple:
140-1221-1229 calculated from 2(35)(2), 35² – 2², 35² + 2²

Here’s another way we know that 1229 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 2² = 1229 with 35 and 2 having no common prime factors, 1229 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1229 ≈ 35.1. Since 1229 is not divisible by 5, 13, 17, or 29, we know that 1229 is a prime number.

 

1223 and Level 3

If you’ve been too anxious to try solving a level 3 puzzle in the past, you have no excuse for not trying this one. This might be the easiest level 3 puzzle I’ve ever published. Just write the factors for 40 and 48 in the proper cells, then work your way down the puzzle writing only numbers from 1 to 10 in the first column and the top row. Seriously, you can do this one!

Print the puzzles or type the solution in this excel file: 10-factors-1221-1231

Now I’ll write a little about the number 1223:

  • 1223 is a prime number.
  • Prime factorization: 1223 is prime.
  • The exponent of prime number 1223 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1223 has exactly 2 factors.
  • Factors of 1223: 1, 1223
  • Factor pairs: 1223 = 1 × 1223
  • 1223 has no square factors that allow its square root to be simplified. √1223 ≈ 34.97142

How do we know that 1223 is a prime number? If 1223 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1223 ≈ 34.97. Since 1223 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1223 is a prime number.

1223 is the sum of the twenty-one prime numbers from 17 to 103.