1177 and Level 3

Why are two of the clues in today’s level three puzzle in red?

You still figure out the common factor of 32 and 72, then work down the first column cell by cell filling in factors as you go, BUT you won’t be able to know what factors to use for 9 unless you look at the number 15 first. You don’t have a problem with that, do you?

Print the puzzles or type the solution in this excel file: 10-factors-1174-1186

Now I’ll tell you some facts about the number 1177:

  • 1177 is a composite number.
  • Prime factorization: 1177 = 11 × 107
  • 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 1177 has exactly 4 factors.
  • Factors of 1177: 1, 11, 107, 1177
  • Factor pairs: 1177 = 1 × 1177 or 11 × 107
  • 1177 has no square factors that allow its square root to be simplified. √1177 ≈ 34.30743

1177 is a palindrome in two bases:
It’s 414 in BASE 17 because 4(17²) + 1(17) + 4(1) = 1177
and 1E1 in BASE 28 (E is 14 base 10) because 28² + 14(28) + 1 = 1177

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1165 and Level 3

Just because you start with the clues at the top of the puzzle and work down cell by cell to solve a level 3 puzzle doesn’t mean that you won’t have to do any thinking. Believe me, you will still have to THINK to solve this puzzle!

Print the puzzles or type the solution in this excel file: 12 factors 1161-1173

Now I’ll write a little bit about the number 1165:

  • 1165 is a composite number.
  • Prime factorization: 1165 = 5 × 233
  • 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 1165 has exactly 4 factors.
  • Factors of 1165: 1, 5, 233, 1165
  • Factor pairs: 1165 = 1 × 1165 or 5 × 233
  • 1165 has no square factors that allow its square root to be simplified. √1165 ≈ 34.132096

34² + 3² = 1165
29² + 18² = 1165

1165 is the hypotenuse of FOUR Pythagorean triples:
204-1147-1165 calculated from 2(34)(3), 34² – 3², 34² + 3²
517-1044-1165 calculated from 29² – 18², 2(29)(18), 29² + 18²
525-1040-1165 which is 5 times (105-208-233)
699-932-1165 which is (3-4-5) times 233

1151 and Level 3

If you know the common prime factor for 27 and 30, then you can at least start this puzzle. If you work down the first column cell by cell using logic, you should be able to solve the puzzle, too. Good luck!

Print the puzzles or type the solution in this excel file: 10-factors-1148-1160

Here are some facts about the number 1151:

  • 1151 is a prime number.
  • Prime factorization: 1151 is prime.
  • The exponent of prime number 1151 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1151 has exactly 2 factors.
  • Factors of 1151: 1, 1151
  • Factor pairs: 1151 = 1 × 1151
  • 1151 has no square factors that allow its square root to be simplified. √1151 ≈ 33.92639

How do we know that 1151 is a prime number? If 1151 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1151 ≈ 33.9. Since 1151 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1151 is a prime number.

1151 is the sum of consecutive primes three different ways:
It is the sum of the twenty-three prime numbers from 7 to 101.
223 + 227 + 229 + 233 + 239 = 1151 and
379 + 383 + 389 = 1151

Stetson.edu states that 1151 is the smallest number that is the sum of consecutive prime numbers four different ways, I think they must be considering 1151 = 1151 to be one of those ways.

1151 is palindrome 1L1 in BASE 25 (L is 21 base 10)
because 25² + 21(25) + 1 = 1151

1138 and Level 3

There are two common factors of 10 and 14, but only one of them will put only numbers from 1 to 12 in the first column. Do you know what that factor is? If you do, figure out where to put the factors of 22, 66, 15 and so forth to make this puzzle function like a multiplication table. Each number from 1 to 12 can only appear once in the first column and once in the top row. You can do this!

Print the puzzles or type the solution in this excel file: 12 factors 1134-1147

Now I’d like to share some facts about the number 1138:

  • 1138 is a composite number.
  • Prime factorization: 1138 = 2 × 569
  • 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 1138 has exactly 4 factors.
  • Factors of 1138: 1, 2, 569, 1138
  • Factor pairs: 1138 = 1 × 1138 or 2 × 569
  • 1138 has no square factors that allow its square root to be simplified. √1138 ≈ 33.73426

33² + 7² = 1138

1138 is the hypotenuse of a Pythagorean triple:
462-1040-1138 calculated from 2(33)(7), 33² – 7², 33² + 7²

1124 and Level 3

The common factors of 54 and 60 are 1, 2, 3, and 6. Just one of those common factors will put only numbers from 1 to 10 in the top row. That’s the factor you need to choose. To complete the puzzle, all the numbers from 1 to 10 must go in both the first column and the top row. Can you solve this puzzle?

Print the puzzles or type the solution in this excel file: 10-factors-1121-1133

Here are a few facts about the number 1124:

  • 1124 is a composite number.
  • Prime factorization: 1124 = 2 × 2 × 281, which can be written 1124 = 2² × 281
  • 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 1124 has exactly 6 factors.
  • Factors of 1124: 1, 2, 4, 281, 562, 1124
  • Factor pairs: 1124 = 1 × 1124, 2 × 562, or 4 × 281
  • Taking the factor pair with the largest square number factor, we get √1124 = (√4)(√281) = 2√281 ≈ 33.52611

1124 is the hypotenuse of a Pythagorean triple:
640-924-1124 which is 4 times 160-231-281

If I asked you to tell me what is significant about this set of numbers {13, 16, 19, 22}, what would you say?

Perhaps you would tell me they make an arithmetic sequence in which the common difference is 3.

What you probably wouldn’t tell me is that 1124 is a palindrome in those four bases!
It’s 686 in BASE 13 because 6(13²) + 8(13) + 6(1) = 1124,
464 in BASE 16 because 4(16²) + 6(16) + 4(1) = 1124
323 in BASE 19 because 3(19²) + 2(19) + 3(1) = 1124, and
272 in BASE 22 because 2(22²) + 7(22) + 2(1) = 1124

1115 and Level 3

You will have to know the 11 and 12 times tables to solve this Level 3 Find the Factors 1-12 puzzle, but I’m sure you can do it! Stick with it, and don’t give up! Start with the clues at the top of the puzzle and work down row by row until it’s completed.

Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

1115 is not a clue just the puzzle number. Here are some facts about the number 1115:

  • 1115 is a composite number.
  • Prime factorization: 1115 = 5 × 223
  • 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 1115 has exactly 4 factors.
  • Factors of 1115: 1, 5, 223, 1115
  • Factor pairs: 1115 = 1 × 1115 or 5 × 223
  • 1115 has no square factors that allow its square root to be simplified. √1115 ≈ 33.39162

1115 is the sum of nine consecutive prime numbers:
103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 = 1115

1115 is the hypotenuse of a Pythagorean triple:
669-892-1115 which is (3-4-5) times 223

1115 is palindrome 2B2 in BASE 21 (B is 11 base 10)
because 2(21²) + 11(21) + 2(1) = 1115

 

1104 and Level 3

If this were a Find the Factors 1-12 puzzle, the possible common factors for 12 and 48 would be 4, 6, and 12. But we can only have factors from 1 to 10 so only one of those common factors will work with this puzzle. If you know which one, you are well on your way to solving it.

Print the puzzles or type the solution in this excel file: 10-factors-1102-1110

Here are some facts about the number 1104:

  • 1104 is a composite number.
  • Prime factorization: 1104 = 2 × 2 × 2 × 2 × 3 × 23, which can be written 1104 = 2⁴ × 3 × 23
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1104 has exactly 20 factors.
  • Factors of 1104: 1, 2, 3, 4, 6, 8, 12, 16, 23, 24, 46, 48, 69, 92, 138, 184, 276, 368, 552, 1104
  • Factor pairs: 1104 = 1 × 1104, 2 × 552, 3 × 368, 4 × 276, 6 × 184, 8 × 138, 12 × 92, 16 × 69, 23 × 48, or 24 × 46
  • Taking the factor pair with the largest square number factor, we get √1104 = (√16)(√69) = 4√69 ≈ 33.2265.

1104 is the sum of the sixteen prime numbers from 37 to 103. Do you know what those prime numbers are?

1104 is also the sum of eight consecutive primes and two consecutive primes:
113 + 127 + 131 + 137 + 139 + 149 +151 + 157  = 1104
547 + 557 = 1104

1097 and Level 3

72 and 27 are mirror images of each other. What is the largest number that will divide evenly into both of them? Put the answer to that question under the x, and you will have completed the first step in solving this multiplication table puzzle.

Print the puzzles or type the solution in this excel file: 12 factors 1095-1101

Here’s a little bit more about the number 1097:

  • 1097 is a prime number.
  • Prime factorization: 1097 is prime.
  • The exponent of prime number 1097 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1097 has exactly 2 factors.
  • Factors of 1097: 1, 1097
  • Factor pairs: 1097 = 1 × 1097
  • 1097 has no square factors that allow its square root to be simplified. √1097 ≈ 33.12099

How do we know that 1097 is a prime number? If 1097 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1097 ≈ 33.1. Since 1097 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1097 is a prime number.

1097 is the final prime number in the prime triplet, 1091-1093-1097.

1097 is the sum of two squares:
29² + 16² = 1097

1097 is the hypotenuse of a primitive Pythagorean triple:
585-928-1097 calculated from 29² – 16², 2(29)(16), 29² + 16²

Here’s another way we know that 1097 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 29² + 16² = 1097 with 29 and 16 having no common prime factors, 1097 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1097 ≈ 33.1. Since 1097 is not divisible by 5, 13, 17, or 29, we know that 1097 is a prime number.

 

 

1087 and Level 3

Using logic, start with the clue on the top row and work yourself down row by row filling in the appropriate factors while you go. You might find this level 3 puzzle a little tricky near the bottom of the puzzle, so I didn’t want to wait to share it with you. Happy factoring!

Print the puzzles or type the solution in this excel file: 10-factors-1087-1094

1087 is the first prime since 1069, which was 18 numbers ago! What else can I tell you about it?

  • 1087 is a prime number.
  • Prime factorization: 1087 is prime.
  • The exponent of prime number 1087 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1087 has exactly 2 factors.
  • Factors of 1087: 1, 1087
  • Factor pairs: 1087 = 1 × 1087
  • 1087 has no square factors that allow its square root to be simplified. √1087 ≈ 32.96968

How do we know that 1087 is a prime number? If 1087 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1087 ≈ 33. Since 1087 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1087 is a prime number.

1087 is also palindrome 767 in BASE 12 because
7(12²) + 6(12) + 7(1) = 1087

1083 and Level 3

Start with the two clues near the top of this level 3 puzzle. Find their common factor that will put only numbers from 1 to 12 in the top row. Then work down the puzzle row by row filling in factors from 1 to 12 as you go. It won’t take you long to complete this puzzle!

Print the puzzles or type the solution in this excel file: 12 factors 1080-1086

1 + 0 + 8 + 3 = 12, so 1083 can be evenly divided by 3. What else can I tell you about that number?

  • 1083 is a composite number.
  • Prime factorization: 1083 = 3 × 19 × 19, which can be written 1083 = 3 × 19²
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 × 3  = 6. Therefore 1083 has exactly 6 factors.
  • Factors of 1083: 1, 3, 19, 57, 361, 1083
  • Factor pairs: 1083 = 1 × 1083, 3 × 361, or 19 × 57
  • Taking the factor pair with the largest square number factor, we get √1083 = (√361)(√3) = 19√3 ≈ 32.90897

1083 looks interesting when it is written in some other bases:
It’s 575 in BASE 14 because 5(14²) + 7(14) + 5(1) = 1083,
363 in BASE 18 because 3(18²) + 6(18) + 3(1) = 3(18² + 36 + 1) = 3(361) = 1083,
300 in BASE 19 because 3(19²) = 1083, and
it’s 212 in BASE 23 because 2(23²) + 1(23) + 2(1) = 1083