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.

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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.

 

1217 Squeegee Mystery

The murder weapon for today’s mystery appears to be a squeegee! Study all the clues in the puzzle and see if you can find all the needed factors to solve this mystery. As in every whodunnit, there is only one true solution.

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Now here are a few facts about the number 1217:

  • 1217 is a prime number.
  • Prime factorization: 1217 is prime.
  • The exponent of prime number 1217 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1217 has exactly 2 factors.
  • Factors of 1217: 1, 1217
  • Factor pairs: 1217 = 1 × 1217
  • 1217 has no square factors that allow its square root to be simplified. √1217 ≈ 34.88553

How do we know that 1217 is a prime number? If 1217 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1217 ≈ 34.9. Since 1217 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1217 is a prime number.

31² + 16² = 1217

1217 is the hypotenuse of a Pythagorean triple:
705-992-1217 calculated from 31² – 16², 2(31)(16), 31² + 16²

Here’s another way we know that 1217 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 16² = 1217 with 31 and 16 having no common prime factors, 1217 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1217 ≈ 34.9. Since 1217 is not divisible by 5, 13, 17, or 29, we know that 1217 is a prime number.

1213 and Level 3

This puzzle would be a lot tougher to solve if I didn’t put the clues in the order that I did. Just start at the top of the puzzle and work your way down the puzzle clue by clue until you get to the bottom of the puzzle and have the entire thing solved.

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Now I’ll share a few facts about the number 1213:

  • 1213 is a prime number.
  • Prime factorization: 1213 is prime.
  • The exponent of prime number 1213 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1213 has exactly 2 factors.
  • Factors of 1213: 1, 1213
  • Factor pairs: 1213 = 1 × 1213
  • 1213 has no square factors that allow its square root to be simplified. √1213 ≈ 34.82815

How do we know that 1213 is a prime number? If 1213 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1213 ≈ 34.8. Since 1213 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1213 is a prime number.

1213 is the sum of these nine consecutive prime numbers:
109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 = 1213

27² + 22² = 1213
1213 is the hypotenuse of a Pythagorean triple:
245-1188-1213 calculated from 27² – 22², 2(27)(22), 27² + 22²

Here’s another way we know that 1213 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 27² + 22² = 1213 with 27 and 22 having no common prime factors, 1213 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1213 ≈ 34.8. Since 1213 is not divisible by 5, 13, 17, or 29, we know that 1213 is a prime number.

 

1193 Math Carnival Games

During the last week of every month, there is a math education blog carnival happening somewhere in the blogosphere. This month it will happen on my blog! Why do I get to host it? I sent an email to Denise Gaskins who coordinates the carnival and requested the privilege. If you would like to host it in the future, let her know.

In the meantime, you can help me with my carnival. Math can be so much fun for kids from preschool age and even all the way up to high school. If you blog about that, I would love to include one or more of your posts in my carnival. You’ve poured your heart and soul into your posts, so why not promote it at no cost to you?  Don’t be shy! I want to read it, and other people will want to read it, too.

The deadline for submitting posts to my carnival is Friday, August 24th. There is a form for you to submit a link to your post on Denise Gaskins website. Then the following week you will be able to enjoy the carnival even more because of your participation!

Now it will be my pleasure to tell you a few facts about the number 1193:

  • 1193 is a prime number.
  • Prime factorization: 1193 is prime.
  • The exponent of prime number 1193 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1193 has exactly 2 factors.
  • Factors of 1193: 1, 1193
  • Factor pairs: 1193 = 1 × 1193
  • 1193 has no square factors that allow its square root to be simplified. √1193 ≈ 34.53983

How do we know that 1193 is a prime number? If 1193 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1193 ≈ 34.5. Since 1193 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1193 is a prime number.

1193 is the sum of five consecutive prime numbers:
229 + 233 + 239 + 241 + 251 = 1193

32² + 13² = 1193

1193 is the hypotenuse of a Pythagorean triple:
832-855-1193 calculated from 2(32)(13), 32² – 13², 32² + 13²

Here’s another way we know that 1193 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 32² + 13² = 1193 with 32 and 13 having no common prime factors, 1193 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1193 ≈ 34.5. Since 1193 is not divisible by 5, 13, 17, or 29, we know that 1193 is a prime number.

1187 and Level 1

What is the biggest number that can divide all the clues in today’s puzzle without leaving a remainder? If you can answer that question, then you also know the greatest common factor of all those clues. It really is that simple. You can solve this puzzle!

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Now I’ll share some information about the number 1187:

  • 1187 is a prime number.
  • Prime factorization: 1187 is prime.
  • The exponent of prime number 1187 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1187 has exactly 2 factors.
  • Factors of 1187: 1, 1187
  • Factor pairs: 1187 = 1 × 1187
  • 1187 has no square factors that allow its square root to be simplified. √1187 ≈ 34.45287

How do we know that 1187 is a prime number? If 1187 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1187 ≈ 34.5. Since 1187 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1187 is a prime number.

1187 is the sum of the nineteen prime numbers from 23 to 103.
It is also the sum of three consecutive primes:
389 + 397 + 401 = 1187