1304 and Level 3

Which factor pairs of 45 and 18 have only numbers from 1 to 10 in them? Answer that question, put the factors in the appropriate places, and then work your way down this level 3 puzzle cell by cell until you’ve solved it.

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

Here are a few facts about the number 1304:

  • 1304 is a composite number.
  • Prime factorization: 1304 = 2 × 2 × 2 × 163, which can be written 1304 = 2³ × 163
  • 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 1304 has exactly 8 factors.
  • Factors of 1304: 1, 2, 4, 8, 163, 326, 652, 1304
  • Factor pairs: 1304 = 1 × 1304, 2 × 652, 4 × 326, or 8 × 163
  • Taking the factor pair with the largest square number factor, we get √1304 = (√4)(√326) = 2√326 ≈ 36.11094

1304 is the difference of two squares two ways:
327² – 325² = 1304
165² – 161² = 1304

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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

 

1283 and Level 3

What is the greatest common factor of 28 and 63? If you know, then you can probably figure out this puzzle!

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

Here are some facts about the number 1283:

  • 1283 is a prime number.
  • Prime factorization: 1283 is prime.
  • The exponent of prime number 1283 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1283 has exactly 2 factors.
  • Factors of 1283: 1, 1283
  • Factor pairs: 1283 = 1 × 1283
  • 1283 has no square factors that allow its square root to be simplified. √1283 ≈ 35.81899

How do we know that 1283 is a prime number? If 1283 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1283 ≈ 35.8. Since 1283 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1283 is a prime number.

1283 is the sum of the seventeen prime numbers from 41 to 109,
AND it is the sum of the thirteen primes from 71 to 131.

1273 and Level 3

What’s the only common factor or 12 and 11? Write factors of 12 and 11 where they belong on the puzzle below. Then starting back up at the top of the puzzle, go down the puzzle writing the appropriate factors cell by cell until you’re done. You can do this!

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

Here’s some information about the number 1273:

  • 1273 is a composite number.
  • Prime factorization: 1273 = 19 × 67
  • 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 1273 has exactly 4 factors.
  • Factors of 1273: 1, 19, 67, 1273
  • Factor pairs: 1273 = 1 × 1273 or 19 × 67
  • 1273 has no square factors that allow its square root to be simplified. √1273 ≈ 35.67913

1273 is also palindrome 10011111001 in BASE 2.

 

1254 and Level 3

Find the common factor of 8 and 80 so that only numbers from 1 to 12 will be put in the top row of this multiplication table puzzle. Then work down row by row writing the factors of each clue so that the numbers from 1 to 12 appear only once in both the first column and the top row. You can do this!

Print the puzzles or type the solution in this excel file: 12 factors 1251-1258

Here are a few facts about the post number, 1254:

  • 1254 is a composite number.
  • Prime factorization: 1254 = 2 × 3 × 11 × 19
  • 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 1254 has exactly 16 factors.
  • Factors of 1254: 1, 2, 3, 6, 11, 19, 22, 33, 38, 57, 66, 114, 209, 418, 627, 1254
  • Factor pairs: 1254 = 1 × 1254, 2 × 627, 3 × 418, 6 × 209, 11 × 114, 19 × 66, 22 × 57, or 33 × 38
  • 1254 has no square factors that allow its square root to be simplified. √1254 ≈ 35.41186

1254 is the sum of the twenty-four prime numbers from 7 to 103. Do you know what those prime numbers are?

1245 and Level 3

If you know the greatest common factor of 15 and 20, then you can begin to solve this puzzle. Since this is a level 3 puzzle, look at the clues starting at the top of the puzzle and work your way down, writing in the factors as you go. You can do this!

Print the puzzles or type the solution in this excel file: 10-factors-1242-1250

Now I’ll share some facts about the number 1245:

  • 1245 is a composite number.
  • Prime factorization: 1245 = 3 × 5 × 83
  • 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 1245 has exactly 8 factors.
  • Factors of 1245: 1, 3, 5, 15, 83, 249, 415, 1245
  • Factor pairs: 1245 = 1 × 1245, 3 × 415, 5 × 249, or 15 × 83
  • 1245 has no square factors that allow its square root to be simplified. √1245 ≈ 35.28456

1245 is also the hypotenuse of a Pythagorean triple:
747-996-1245 which is (3-4-5) times 249

1235 and Level 3

Do you know the greatest common factor of 28 and 35? If you do, then you can solve this puzzle by writing each number from 1 to 12 in both the first column and the top row. Since this is a level 3 puzzle, you can begin with the clues at the top of the puzzle and work your way down cell by cell. Good luck!

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

Now I’ll share some facts about the number 1235:

  • 1235 is a composite number.
  • Prime factorization: 1235 = 5 × 13 × 19
  • 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 1235 has exactly 8 factors.
  • Factors of 1235: 1, 5, 13, 19, 65, 95, 247, 1235
  • Factor pairs: 1235 = 1 × 1235, 5 × 247, 13 × 95, or 19 × 65
  • 1235 has no square factors that allow its square root to be simplified. √1235 ≈ 35.14257

1235 is the hypotenuse of FOUR Pythagorean triples:
304-1197-1235 which is 19 times (16-63-65)
741-988-1235 which is (3-4-5) times 247
627-1064-1235 which is 19 times (33-56-65)
475-1140-1235 which is (5-12-13) times 95

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.

 

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.

 

1203 and Level 3

At the top of this level 3 puzzle are two clues that will tell you where to put three of the factors needed to solve the puzzle. After you find those three clues work down looking at the clues cell by cell until you have the entire puzzle solved.

Print the puzzles or type the solution in this excel file: 10-factors-1199-1210

Here are a few facts about the number 1203:

  • 1203 is a composite number.
  • Prime factorization: 1203 = 3 × 401
  • 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 1203 has exactly 4 factors.
  • Factors of 1203: 1, 3, 401, 1203
  • Factor pairs: 1203 = 1 × 1203 or 3 × 401
  • 1203 has no square factors that allow its square root to be simplified. √1203 ≈ 34.68429

Since 1203 is only made from three consecutive numbers (1, 2, 3) and zeros, it has to be divisible by 3.

1203 is the hypotenuse of a Pythagorean triple:
120-1197-1203 which is 3(40-399-401)