1139 and Level 4

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The common factors of 30 and 48 are 1, 2, 3, and 6. The rules of a Find the Factors 1 – 12 puzzle require that only numbers from 1 to 12 go in either the first column or the top row. Each number can be used only once in each place. Which common factor of 30 and 48 must be chosen?

Here are some facts about the number 1139:

  • 1139 is a composite number.
  • Prime factorization: 1139 = 17 × 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 1139 has exactly 4 factors.
  • Factors of 1139: 1, 17, 67, 1139
  • Factor pairs: 1139 = 1 × 1139 or 17 × 67
  • 1139 has no square factors that allow its square root to be simplified. √1139 ≈ 33.74907

1139 is the sum of seven/eleven consecutive primes:
149 + 151 + 157 + 163 + 167 + 173 + 179 = 1139
79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 = 1139

1139 is the hypotenuse of a Pythagorean triple:
536-1005-1139 which is (8-15-17) times 67

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1138 and Level 3

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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!

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

1137 and Level 2

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There’s no reason to let this puzzle tie you up in knots. Solving it will only require you to do a little thinking to make all the factors fall into place. I’m sure you can do it!

Here is a little information about the number 1137:

  • 1137 is a composite number.
  • Prime factorization: 1137 = 3 × 379
  • 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 1137 has exactly 4 factors.
  • Factors of 1137: 1, 3, 379, 1137
  • Factor pairs: 1137 = 1 × 1137 or 3 × 379
  • 1137 has no square factors that allow its square root to be simplified. √1137 ≈ 33.71943

1137 is a palindrome in three different bases:
It’s 10001110001 in BASE 2 because 2¹⁰ + 2⁶ + 2⁵ + 2⁴ + 2⁰ = 1137,
696 in BASE 13 because 6(13²) + 9(13) + 6(1) = 1137, and
393 in BASE 18 because 3(18²) + 9(18) + 3(1) = 1137

1136 and Level 1

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This puzzle has 20 clues to help you to know where to write the factors 1 to 12 in both the first column and the top row. After you find all the factors you can make the puzzle be a special type of multiplication table.

Here are a few facts about the number 1136:

  • 1136 is a composite number.
  • Prime factorization: 1136 = 2 × 2 × 2 × 2 × 71, which can be written 1136 = 2⁴ × 71
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1136 has exactly 10 factors.
  • Factors of 1136: 1, 2, 4, 8, 16, 71, 142, 284, 568, 1136
  • Factor pairs: 1136 = 1 × 1136, 2 × 568, 4 × 284, 8 × 142, or 16 × 71
  • Taking the factor pair with the largest square number factor, we get √1136 = (√16)(√71) = 4√71 ≈ 33.7046

1136 is palindrome 2C2 in BASE 21 (C is 12 base 10)
because 2(21²) + 12(21) + 2(1) = 1136

1135 is a Centered Triangular Number

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The reason 1135 is a centered triangular number is illustrated in this graphic:

(26×27 + 27×28 + 28×29)/2 = 1135 means that 1135 is the sum of the 26th, 27th, and 28th triangular numbers. The sum of three consecutive triangular numbers is always a centered triangular number.

Here are some more facts about the number 1135:

  • 1135 is a composite number.
  • Prime factorization: 1135 = 5 × 227
  • 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 1135 has exactly 4 factors.
  • Factors of 1135: 1, 5, 227, 1135
  • Factor pairs: 1135 = 1 × 1135 or 5 × 227
  • 1135 has no square factors that allow its square root to be simplified. √1135 ≈ 33.68976

1135 is the sum of the seventeen prime numbers from 31 to 103.
It is also the sum of three consecutive prime numbers:
373 + 379 + 383 = 1135

1135 is the hypotenuse of a Pythagorean triple:
681-908-1135 which is (3-4-5) times 227

1135 is a palindrome in two different bases:
It’s 7A7 in BASE 12 (A is 10 base 10) because 7(12²) + 10(12) + 7(1) = 1135,
and 1F1 in BASE 27 (F is 15 Base 10) because 27² + 15(27) + 1 = 1135

1134 Some Likely Factor Trees

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I could easily make nine factor trees for 1134 using different factor pairs of 1134. However, most people aren’t likely to begin a factor tree by first dividing 1134 by 14, 18, 21, or 27. Most people will want to start with a 1-digit divisor of 1134.

Most people will also use only one color of ink or pencil lead to make a factor tree.

Here are three of 1134’s factor trees that are much more likely to be made by the average person.

How easy is it to find all the prime factors in those one-color trees? You will probably think it is easier for some trees than others.

I think the tree in the middle is the easiest to read. All the prime factors are in the correct order and easy to distinguish from the composite factors in the tree. Although it is similar to using the cake method, I still like the cake method better.

Here are some facts about the number 1134:

  • 1134 is a composite number.
  • Prime factorization: 1134 = 2 × 3 × 3 × 3 × 3 × 7, which can be written 1134 = 2 × 3⁴ × 7
  • The exponents in the prime factorization are 1, 4 and 1. Adding one to each and multiplying we get (1 + 1)(4 + 1)(1 + 1) = 2 × 5 × 2 = 20. Therefore 1134 has exactly 20 factors.
  • Factors of 1134: 1, 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 81, 126, 162, 189, 378, 567, 1134
  • Factor pairs: 1134 = 1 × 1134, 2 × 567, 3 × 378, 6 × 189, 7 × 162, 9 × 126, 14 × 81, 18 × 63, 21 × 54, or 27 × 42
  • Taking the factor pair with the largest square number factor, we get √1134 = (√81)(√14) = 9√14 ≈ 33.67492

1134 is the sum of four consecutive prime numbers:
277 + 281 + 283 + 293 = 1134

1133 A Challenge Puzzle

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I haven’t published a Find the Factors 1 – 10 Challenge puzzle for a while. Getting started on this one shouldn’t be difficult. The challenge will be in finishing it! Don’t guess and check. It can all be done using logic.

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

Here are some facts about the number 1133:

Since both 11 and 33 are divisible by 11, we know 1133 can be evenly divided by 11.

  • 1133 is a composite number.
  • Prime factorization: 1133 = 11 × 103
  • 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 1133 has exactly 4 factors.
  • Factors of 1133: 1, 11, 103, 1133
  • Factor pairs: 1133 = 1 × 1133 or 11 × 103
  • 1133 has no square factors that allow its square root to be simplified. √1133 ≈ 33.66007

1133 is the sum of the twenty-one prime numbers from 13 to 101.
1133 is also the sum of the fifteen prime numbers from 43 to 107.

1132 Mystery

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There are ten simple clues in this Mystery Level puzzle. That’s all that is needed to come up with the unique solution. Can you find it?

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

Now I’ll share some facts about the number 1132:

  • 1132 is a composite number.
  • Prime factorization: 1132 = 2 × 2 × 283, which can be written 1132 = 2² × 283
  • 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 1132 has exactly 6 factors.
  • Factors of 1132: 1, 2, 4, 283, 566, 1132
  • Factor pairs: 1132 = 1 × 1132, 2 × 566, or 4 × 283
  • Taking the factor pair with the largest square number factor, we get √1132 = (√4)(√283) = 2√283 ≈ 33.64521

1132 is the sum of two consecutive prime numbers:
563 + 569 = 1132

1132 is palindrome 1A1 in BASE 29 (A is 10 base 10) because
29² + 10(29) + 1 = 1132

1131 Mystery

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You can’t know if this puzzle is easy or difficult by just looking at it. Give it a try. You might be surprised by your ability to do this puzzle!

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

Here are some facts about the number 1131:

  • 1131 is a composite number.
  • Prime factorization: 1131 = 3 × 13 × 29
  • 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 1131 has exactly 8 factors.
  • Factors of 1131: 1, 3, 13, 29, 39, 87, 377, 1131
  • Factor pairs: 1131 = 1 × 1131, 3 × 377, 13 × 87, or 29 × 39
  • 1131 has no square factors that allow its square root to be simplified. √1131 ≈ 33.63034

1131 is the hypotenuse of FOUR Pythagorean triples:
405-1056-1131 which is 3 times (135-352-377)
435-1044-1131 which is (5-12-13) times 87
456-1035-1131 which is 3 times (152-345-377)
780-819-1131 which is (20-21-29) times 39

1131 looks interesting in a couple of other bases:
It’s 939 in BASE 11 because 9(11²) + 3(11) + 9(1) = 1131
234 in BASE 23 because 2(23²) + 3(23) + 4(1) = 1131

1130 and Level 6

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Today’s puzzle has four sets of clues with two or more common factors. Which common factor should you choose in each case? Guessing and checking will only frustrate you. Study the puzzle instead. The clues work together to give ONE logical solution.

Now here is some information about the number 1130:

  • 1130 is a composite number.
  • Prime factorization: 1130 = 2 × 5 × 113
  • 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 1130 has exactly 8 factors.
  • Factors of 1130: 1, 2, 5, 10, 113, 226, 565, 1130
  • Factor pairs: 1130 = 1 × 1130, 2 × 565, 5 × 226, or 10 × 113
  • 1130 has no square factors that allow its square root to be simplified. √1130 ≈ 33.61547

1130 is the hypotenuse of FOUR Pythagorean triples:
150-1120-1130 which is 10 times (15-112-113)
552-986-1130 which is 2 times (276-493-565)
678-904-1130 which is (3-4-5) times 226
792-806-1130 which is 2 times (396-403-565)

1130 is also a leg in some Pythagorean triples including
1130-12744-12794 calculated from 2(113)(5), 113² – 5², 113² + 5²

1130 is palindrome 505 in BASE 15 because 5(15²) + 5(1) = 5(226) = 1130