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.

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

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!

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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

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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

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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.

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

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

/ ivasallay / Leave a comment

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!

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

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?

/ ivasallay / Leave a comment

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

1124 and Level 3

/ ivasallay / Leave a comment

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

1123 and Level 2

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All the clues in one of the rows of this Level 2 puzzle are prime numbers. The only common factor they have is 1. That fact will get you started with this puzzle which I am sure you can complete if you just give it a try!

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

1123 is also a prime number. Here are some facts about it.

  • 1123 is a prime number.
  • Prime factorization: 1123 is prime.
  • The exponent of prime number 1123 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1123 has exactly 2 factors.
  • Factors of 1123: 1, 1123
  • Factor pairs: 1123 = 1 × 1123
  • 1123 has no square factors that allow its square root to be simplified. √1123 ≈ 33.51119

How do we know that 1123 is a prime number? If 1123 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1123 ≈ 33.5. Since 1123 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1123 is a prime number.

OEIS.org reminds us that 1, 1, 2, 3 are the first four numbers in the Fibonacci sequence.

1123 is the sum of five consecutive prime numbers:
211 + 223 + 227 + 229 + 233 = 1123

1123 is palindrome 797 in BASE 12 because 7(12²) + 9(12) + 7(1) = 1123, and
it’s repdigit 111 in BASE 33 because 33² + 33 + 1 = 1123