1134 Some Likely Factor Trees

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

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1133 A Challenge Puzzle

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

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

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

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

1129 is the Last Prime Number for a While!

The last prime number before 1129 was 1123.

The next prime number after 1129 will be 1151.

1151 – 1129 = 22

That’s the largest gap between primes so far! The previous record was 20. The chart below shows the gaps between all the prime numbers up to 1163.

Here’s more about the number 1129:

  • 1129 is a prime number.
  • Prime factorization: 1129 is prime.
  • The exponent of prime number 1129 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1129 has exactly 2 factors.
  • Factors of 1129: 1, 1129
  • Factor pairs: 1129 = 1 × 1129
  • 1129 has no square factors that allow its square root to be simplified. √1129 ≈ 33.6006

How do we know that 1129 is a prime number? If 1129 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1129 ≈ 33.6. Since 1129 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1129 is a prime number.

27² + 20²  = 1129

1029 is the hypotenuse of a Pythagorean triple:
329-1080-1129 calculated from 27² – 20², 2(27)(20), 27² + 20²

Here’s another way we know that 1129 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 27² + 20² = 1129 with 26 and 21 having no common prime factors, 1129 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1129 ≈ 33.6. Since 1129 is not divisible by 5, 13, 17, or 29, we know that 1129 is a prime number.

1129 is also palindrome 1N1 in BASE 24 (N is 23 base 10)
because 24² + 23(24) + 1 = 1129

1128 is the 24th Hexagonal Number

1128 is the 24th hexagonal number because of the way that it can be factored:
2(24²) – 24 = 1128,
(2(24) – 1)24 = 1128
or simply 47(24) = 1128.

This is what the 24th hexagonal number looks like when it is made with 1128 tiny squares arranged into a hexagon:

All hexagonal numbers are also triangular numbers. 1128 is the 47th triangular number:

What else can the factors of 1128 tell us?

  • 1128 is a composite number.
  • Prime factorization: 1128 = 2 × 2 × 2 × 3 × 47, which can be written 1128 = 2³ × 3 × 47
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1128 has exactly 16 factors.
  • Factors of 1128: 1, 2, 3, 4, 6, 8, 12, 24, 47, 94, 141, 188, 282, 376, 564, 1128
  • Factor pairs: 1128 = 1 × 1128, 2 × 564, 3 × 376, 4 × 282, 6 × 188, 8 × 141, 12 × 94, or 24 × 47
  • Taking the factor pair with the largest square number factor, we get √1128 = (√4)(√282) = 2√282 ≈ 33.58571

1127 and Level 5

If the clues in this puzzle were in a Find the Factors 1 – 12, puzzle, the needed factors might be completely different than the ones in this puzzle’s solution. Fortunately, we can only use factors from 1 to 10, so this puzzle will make you think, but shouldn’t be so difficult.

Here are a few facts about the number 1127:

  • 1127 is a composite number.
  • Prime factorization: 1127 = 7 × 7 × 23, which can be written 1127 = 7² × 23
  • 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 1127 has exactly 6 factors.
  • Factors of 1127: 1, 7, 23, 49, 161, 1127
  • Factor pairs: 1127 = 1 × 1127, 7 × 161, or 23 × 49
  • Taking the factor pair with the largest square number factor, we get √1127 = (√49)(√23) = 7√23 ≈ 33.57082

1127 is palindrome 5115 in BASE 6 because 5(6³) + 1(6²) + 1(6) + 5(1) = 1127

1126 and Level 4

Since we are only using factors from 1 to 10, we have only one common factor of 12 and 9 to consider instead of two. Also, you will need to ask yourself, “Where is the only place ____ can fit in the first column (or the top row).” to solve this puzzle. Good Luck!

Here are some facts about the number 1126:

  • 1126 is a composite number.
  • Prime factorization: 1126 = 2 × 563
  • 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 1126 has exactly 4 factors.
  • Factors of 1126: 1, 2, 563, 1126
  • Factor pairs: 1126 = 1 × 1126 or 2 × 563
  • 1126 has no square factors that allow its square root to be simplified. √1126 ≈ 33.55592

I chuckled when I noticed that the first five digits of √1126 are all the factors in 1125’s prime factorization.

1126 is palindrome 1K1 in BASE 25 (K is 20 base 10) because 25² + 20(25) + 1 = 1126

1125 May I Interest You In a Little Cake?

1 + 1 + 2 + 5 = 9, so 1125 can be evenly divided by both 3 and 9.
1125 can also be evenly divided by 5 because the last digit can be divided by 5.
1125 can be evenly divided by 25 because the last two digits can be divided by 25.
1125 can be evenly divided by 125 because the last three digits can be divided by 125.

May I interest you in a little cake? Although I’ve put many factor trees on this website, I actually prefer making factor cakes. I like them because all the prime factors can easily be found IN ORDER on the outside of the cake. The more layers the factor cake has, the more I like it. Here is THE factor cake for 1125:

As you can see, the prime factorization of 1125 is 3 × 3 × 5 × 5 × 5. All of its prime factors are written nicely in order. None of them are hiding anyplace like they might in a factor tree.

Here are some facts about the number 1125:

  • 1125 is a composite number.
  • Prime factorization: 1125 = 3 × 3 × 5 × 5 × 5, which can be written 1125 = 3² × 5³
  • The exponents in the prime factorization are 2 and 3. Adding one to each and multiplying we get (2 + 1)(3 + 1) = 3 × 4 = 12. Therefore 1125 has exactly 12 factors.
  • Factors of 1125: 1, 3, 5, 9, 15, 25, 45, 75, 125, 225, 375, 1125
  • Factor pairs: 1125 = 1 × 1125, 3 × 375, 5 × 225, 9 × 125, 15 × 75, or 25 × 45
  • Taking the factor pair with the largest square number factor, we get √1125 = (√225)(√5) = 15√5 ≈ 33.54102

1125 is the hypotenuse of three Pythagorean triples:
675-900-1125 which is (3-4-5) times 225,
315-1080-1125 which is (7-24-25) times 45, and
396-1053-1125 which is 9 times (44-117-125)

1125 looks interesting to me when it is written in these bases:
It’s 5A5 in BASE 14 (A is 10 base 10) because 5(14²) + 10(14) + 5(1) = 1125
500 in BASE 15 because 5(15²) = 1125, and
it’s 3F3 in BASE 17 (F is 15 base 10) because 3(17²) + 15(17) + 3(1) = 1125