Mathematics

1140 is the 18th Tetrahedral Number

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1140 is the 18th tetrahedral number because it satisfies this formula:
(18)(18+1)(18+2)/6 = 1140

It is the 18th tetrahedral number because it is the sum of the first 18 triangular numbers:

Since 18 is an even number, 1140 is the sum of the first 9 EVEN squares.

If the 1140 tiny squares in that graphic were cubes, they could be stacked into a tower with either a triangular base or a square base. Then we would see the beauty of this tetrahedral number.

We can see the number 1140 as well as ALL the previous tetrahedral numbers on this portion of Pascal’s Triangle. (They are the green squares.):

1140 has its place as the 3rd number (as well as the 17th number) on the 20th row of Pascal’s triangle because of this next formula:

Here are some other facts about the number 1140:

  • 1140 is a composite number.
  • Prime factorization: 1140 = 2 × 2 × 3 × 5 × 19, which can be written 1140 = 2² × 3 × 5 × 19
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1140 has exactly 24 factors.
  • Factors of 1140: 1, 2, 3, 4, 5, 6, 10, 12, 15, 19, 20, 30, 38, 57, 60, 76, 95, 114, 190, 228, 285, 380, 570, 1140
  • Factor pairs: 1140 = 1 × 1140, 2 × 570, 3 × 380, 4 × 285, 5 × 228, 6 × 190, 10 × 114, 12 × 95, 15 × 76, 19 × 60, 20 × 57, or 30 × 38,
  • Taking the factor pair with the largest square number factor, we get √1140 = (√4)(√285) = 2√285 ≈ 33.76389

Here are some factor trees that use 11 of 1140’s factor pairs:

1140 is the sum of consecutive prime numbers two different ways:
179 + 181 + 191 + 193 + 197 + 199 = 1140,
569 + 571 = 1140

1140 is the hypotenuse of a Pythagorean triple:
684-912-1140 which is (3-4-5) times 228

1140 looks interesting when it is written in a couple other bases:
It’s palindrome 474 in BASE 16 because 4(16²) + 7(16) + 4(1) = 1140,
and it’s 330 in BASE 19 because 3(19²) + 3(19) = 3(19² + 19) = 3(19)(20) = 1140

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.

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

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

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

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

1106 and Level 4

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Can you use logic to figure out where all the numbers from 1 to 10 need to go in both the first column and the top row so that this puzzle can become a multiplication table? Give it a try. It’s fun!

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

Here is some information about the number 1106:

  • 1106 is a composite number.
  • Prime factorization: 1106 = 2 × 7 × 79
  • 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 1106 has exactly 8 factors.
  • Factors of 1106: 1, 2, 7, 14, 79, 158, 553, 1106
  • Factor pairs: 1106 = 1 × 1106, 2 × 553, 7 × 158, or 14 × 79
  • 1106 has no square factors that allow its square root to be simplified. √1106 ≈ 33.25658

1106 is repdigit 222 in BASE 23 because 2(23² + 23+ 1) = 2(553) = 1106

Mathemagical Properties of 1105

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1105 is the magic sum of a 13 × 13 magic square. Why?
Because 13×13 = 169 and 169×170÷2÷13 = 13×85 = 1105.

If you follow the location of the numbers 1, 2, 3, 4, all the way to 169 in the magic square, you will see the pattern that I used to make that magic square. If you click on 10-factors-1102-1110  and go to the magic squares tab, you can use the same pattern or try another to create an 11 × 11, 13 × 13, or 15 × 15 magic square. The sums on the rows, columns, and diagonals will automatically populate as you write in the numbers so you can verify that you have indeed created a magic square.

1105 tiny squares can be made into a decagon so we say it is a decagonal number:

Those 1105  tiny squares can also be arranged into a centered square:

Why is 1105 the 24th Centered Square Number? Because it is the sum of consecutive square numbers:
24² + 23² = 1105

But that’s not all! 1105 is the smallest number that is the sum of two squares FOUR different ways:

24² + 23² = 1105
31² + 12² = 1105
32² + 9² = 1105
33² + 4² = 1105

1105 is also the smallest number that is the hypotenuse of THIRTEEN different Pythagorean triples. Yes, THIRTEEN! (Seven was the most any previous number has had.) It is also the smallest number to have FOUR of its Pythagorean triplets be primitives (Those four are in blue type.):

47-1104-1105 calculated from 24² – 23², 2(24)(23), 24² + 23²
105-1100-1105 which is 5 times (21-220-221)
169-1092-1105 which is 13 times (13-84-85)
264-1073-1105 calculated from 2(33)(4), 33² – 4², 33² + 4²
272-1071-1105 which is 17 times (16-63-65)
425-1020-1105 which is (5-12-13) times 85
468-1001-1105 which is 13 times (36-77-85)
520-975-1105 which is (8-15-17) times 65
561-952-1105 which is 17 times (33-56-85)
576-943-1105 calculated from 2(32)(9), 32² – 9², 32² + 9²
663-884-1105 which is (3-4-5) times 221
700-855-1105 which is 5 times (140-171-221)
744-817-1105 calculated from 2(31)(12), 31² – 12², 31² + 12²

Why is it the hypotenuse more often than any previous number? Because of its factors! 1105 = 5 × 13 × 17, so it is the smallest number that is the product of THREE different Pythagorean hypotenuses.

It gets 1 triple for each of its three individual factors: 5, 13, 17, 2 triples for each of the three ways the factors can pair up with each other: 65, 85, 221, and four primitive triples for the one way they can all three be together: 1105. Thus it gets 2º×3 + 2¹×3 + 2²×1 = 3 + 6 + 4 = 13 triples.

Speaking of factors, let’s take a look at 1105’s factoring information:

  • 1105 is a composite number.
  • Prime factorization: 1105 = 5 × 13 × 17
  • 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 1105 has exactly 8 factors.
  • Factors of 1105: 1, 5, 13, 17, 65, 85, 221, 1105
  • Factor pairs: 1105 = 1 × 1105, 5 × 221, 13 × 85, or 17 × 65
  • 1105 has no square factors that allow its square root to be simplified. √1105 ≈ 33.24154

1105 is also a palindrome in four different bases, and I also like the way it looks in base 8:
It’s 10001010001 in BASE 2 because 2¹º + 2⁶ + 2⁴ + 2º = 1105,
101101 in BASE 4 because 4⁵ + 4³ + 4² + 4º = 1105,
2121 in BASE 8 because 2(8³) + 1(8²) + 2(8) + 1(1) = 1105,
313 in BASE 19 because 3(19²) + 1(19) + 3(1) = 1105
1M1 in BASE 24 (M is 22 base 10) because 24² + 22(24) + 1 = 1105

Last, but certainly not least, you wouldn’t think 1105 is a prime number, but it is a pseudoprime: the second smallest Carmichael number. Only Carmichael number 561 is smaller than it is.

A Carmichael number is a composite number that behaves like a prime number by giving a false positive to all of certain quick prime number tests:
1105 passes the test p¹¹⁰⁵ Mod 1105 = p for all prime numbers p < 1105. Here is an image of my computer calculator showing 1105 passing the first five tests! Only a prime number should pass all these tests.

1105 is indeed a number with amazing mathemagical properties!

1100 Now THIS Is a Horse Race!

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Most people know that any prime number has exactly 2 factors. Every composite number has more than 2 factors. I’ve made a chart showing the numbers from 1001 to 1100, their prime factorizations, and the amount of factors each of those numbers has.

The fewest factors any of those numbers have is 2 while the most factors any of the numbers have is 32. Which number of factors appears most often? Let’s have a horse race to find out!

Several horses representing different amounts of factors are lined up for the race. Spoiler alert: some of the horses will barely make it out of the gate. THIS will be an exciting horse race. There will be at least one lead change. You won’t know for sure which horse will cross the finish line first until the end. The second place horse will be SO close to winning.

I haven’t made a horse race this exciting since the 601 to 700 Horse Race, so pick your pony, then scroll down and see how your pony does!

Click on the gif below to make the horse race larger.

Factor Horse Race 1100

make science GIFs like this at MakeaGif

How did your pony do? Were you surprised by the results of the race?

Only perfect squares can have an odd number of factors which explains why 9 and 11 barely make it out of the gate. The numbers in pink have square factors so their square roots can be simplified. That describes 41 of the numbers from 1001 to 1100.

I hope this horse race sparks your curiosity about numbers! Each number is fascinating in its own way.

Let me tell you a few things specifically about the number 1100:

  • 1100 is a composite number.
  • Prime factorization: 1100 = 2 × 2 × 5 × 5 × 11, which can be written 1100 = 2² × 5² × 11
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1100 has exactly 18 factors.
  • Factors of 1100: 1, 2, 4, 5, 10, 11, 20, 22, 25, 44, 50, 55, 100, 110, 220, 275, 550, 1100
  • Factor pairs: 1100 = 1 × 1100, 2 × 550, 4 × 275, 5 × 220, 10 × 110, 11 × 100, 20 × 55, 22 × 50 or 25 × 44
  • Taking the factor pair with the largest square number factor, we get √1100 = (√100)(√11) = 10√11 ≈ 33.16625

1100 is the hypotenuse of two Pythagorean triples:
660-880-1100 which is (3-4-5) times 220
308-1056-1100 which is (7-24-25) times 44

It may interest you to know that 1100 is 3131 in BASE 7.
That’s because 3(7³) + 1(7²) + 3(7) + 1( 1) = 1100.

All those fun facts are straight from the horse’s mouth!

1093 is a STAR!

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(12 × 13 × 14)/2 = 1092 and that makes 1093 a STAR!

1092 had more than its fair share of factors forcing 1093 to have only two factors, but that’s okay because 1093 is a beautiful STAR! Why is it a STAR? Because 1093 is one more than 12 times the 13th triangular number. Do you see those 12 triangles in the image above? Each of them has the same number of tiny squares. The yellow square in the center is the plus one that completes the star.

There are some other reasons why 1093 deserves a gold star:

Not every prime number is in a twin prime, but 1091 and 1093 are twin primes. Even fewer are part of a prime triplet, but those twin primes are part of TWO prime triplets: the 31st and the 32nd! That’s because 1087, 1091, 1093, and 1097 are all prime numbers.

  • 1093 is a prime number.
  • Prime factorization: 1093 is prime.
  • The exponent of prime number 1093 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1093 has exactly 2 factors.
  • Factors of 1093: 1, 1093
  • Factor pairs: 1093 = 1 × 1093
  • 1093 has no square factors that allow its square root to be simplified. √1093 ≈ 33.03329

How do we know that 1093 is a prime number? If 1093 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1093 ≈ 33.1. Since 1093 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1093 is a prime number.

1093 is the sum of two squares:
33² + 2² = 1093

It is the hypotenuse of a primitive Pythagorean triple:
132-1085-1093 calculated from 2(33)(2), 33² – 2², 33² + 2²

Here’s another way we know that 1093 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 33² + 2² = 1093 with 33 and 2 having no common prime factors, 1093 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1093 ≈ 33.1. Since 1093 is not divisible by 5, 13, 17, or 29, we know that 1093 is a prime number.

3⁶ + 3⁵ + 3⁴ + 3³ + 3² + 3¹ + 3⁰ = 1093 so 1093 is represented by 1111111 in BASE 3. That also means that 2(1093) + 1 = 3⁷.

1093 is a palindrome in two bases:
1G1 in BASE 26 (G is 16 base 10) because 26² + 16(26) + 1 = 1093, and
1B1 in BASE 28 (B is 11 base 10) because 28² + 11(28) + 1 = 1093

From OEIS.org and Wikipedia, we learn something quite unique about 1093 – that it is the smaller of the two known Wieferich primes, 1093 and 3511.

Wow! 1093 truly is a STAR!