1183 is the 13th Pentagonal Pyramidal Number

 

1183 is the 13th pentagonal pyramidal number. Here’s an attempt to illustrate that fact. (Try to think 3-dimensionally.):

Here are some more facts about the number 1183:

  • 1183 is a composite number.
  • Prime factorization: 1183 = 7 × 13 × 13, which can be written 1183 = 7 × 13²
  • 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 1183 has exactly 6 factors.
  • Factors of 1183: 1, 7, 13, 91, 169, 1183
  • Factor pairs: 1183 = 1 × 1183, 7 × 169, or 13 × 91
  • Taking the factor pair with the largest square number factor, we get √1183 = (√169)(√7) = 13√7 ≈ 34.39476

1183 is the hypotenuse of two Pythagorean triples:
833-840-1183 which is 7 times (119-120-169)
455-1092-1183 which is (5-12-13) times 91

I like 1183 in a couple of different bases:
It’s palindrome 1121211 in BASE 3 because
3⁶ + 3⁵ + 2(3⁴) + 3³ + 2(3²) + 3 + 1 = 1183,
and it’s 700 in BASE 13 because 7(13²) = 1183

 

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1181 and Level 5

Some parts of this puzzle are easier than others, but it all still a lot of fun! Give it a try and enjoy yourself!

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

Now I’ll share some facts about the number 1181:

  • 1181 is a prime number.
  • Prime factorization: 1181 is prime.
  • The exponent of prime number 1181 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1181 has exactly 2 factors.
  • Factors of 1181: 1, 1181
  • Factor pairs: 1181 = 1 × 1181
  • 1181 has no square factors that allow its square root to be simplified. √1181 ≈ 34.36568

How do we know that 1181 is a prime number? If 1181 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1181 ≈ 34.4. Since 1181 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1181 is a prime number.

34² + 5² = 1181

1181 is the hypotenuse of a Pythagorean triple:
340-1131-1181 calculated from 2(34)( 5), 34² – 5², 34² + 5²

Here’s another way we know that 1181 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 34² + 5² = 1181 with 34 and 5 having no common prime factors, 1181 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1181 ≈ 34.4. Since 1181 is not divisible by 5, 13, 17, or 29, we know that 1181 is a prime number.

1181 is also palindrome 353 in BASE 19
because 3(19²) + 5(19) + 3(1) = 1181

1180 How Are They the Same? How Are They Different?

I liked the vintage buggies at Sosto Museum Village so much that I had to snap a picture and share them with you.

How are they the same? How are they different? When you answer those questions, you are doing mathematics!

You could write the answers to those questions in a Venn Diagram. Then it would be more obvious that you are thinking mathematically!

The number 1180 is like some other numbers but different than others. Here are some facts about it:

  • 1180 is a composite number.
  • Prime factorization: 1180 = 2 × 2 × 5 × 59, which can be written 1180 = 2² × 5 × 59
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1180 has exactly 12 factors.
  • Factors of 1180: 1, 2, 4, 5, 10, 20, 59, 118, 236, 295, 590, 1180
  • Factor pairs: 1180 = 1 × 1180, 2 × 590, 4 × 295, 5 × 236, 10 × 118, or 20 × 59
  • Taking the factor pair with the largest square number factor, we get √1180 = (√4)(√295) = 2√295 ≈ 34.35113

1180 is the hypotenuse of a Pythagorean triple:
708-944-1180 which is (3-4-5) times 236

1180 is the sum of two consecutive prime numbers: 587 + 593 = 1180

1180 is palindrome 1551 in BASE 9 because 1(9³) + 5(9²) + 5(9) + 1(1) = 1180

 

 

1178 School Supplies

There is so much to see at the Ópusztaszer National Heritage Park whose location marks the birthplace of the nation of Hungary. I will mention only two of its attractions in this post.

In the rotunda is a huge cyclorama painting by Arpad Feszty depicting the arrival of the Magyars over 1100 years ago. The painting in of itself is amazing, but it is also augmented with artificial landscapes in front of and all the way around the painting, giving it a 3D effect. Photography in the rotunda is forbidden, but there is no way to capture the magnificence of this work of art in a 2-dimensional photo anyway. (Neither do these few words I’ve written.)

The 15-building museum village includes a school. I was pleased to see some of the authentic school supplies from around the turn of the 20th century and before. This first one is a slate students could use not only to write mathematical calculations but also to graph equations or make bar graphs!

This abacus also made me smile.

I would encourage you to visit Ópusztaszer National Heritage Park should you get to visit Hungary, but give yourselves more than the two hours we did to enjoy all it has to offer.

Now I’ll share some information about the number 1178:

  • 1178 is a composite number.
  • Prime factorization: 1178 = 2 × 19 × 31
  • 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 1178 has exactly 8 factors.
  • Factors of 1178: 1, 2, 19, 31, 38, 62, 589, 1178
  • Factor pairs: 1178 = 1 × 1178, 2 × 589, 19 × 62, or 31 × 38
  • 1178 has no square factors that allow its square root to be simplified. √1178 ≈ 34.322

1178 is a leg in a few Pythagorean triples including
600-1178-1322 calculated from  31² – 19², 2(31)(19), 31² + 19²

1178 is palindrome 212 in BASE 24 because 2(24²) + 1(24) + 2(1) = 1178

1164 Mathematics at Corvin Castle

In the forests of Transylvania in what is now Hunedoara, Romania, a large, beautiful, well-restored, and fascinating castle awaits. At Corvin Castle, you can see how enemies were tortured and royal friends were entertained in the days of kings and knights. However, the room that intrigued me the most was full of mathematics. I didn’t take a picture of every geometric shape that graced its walls; I only took a few. You will have to go there yourself to see all the wonderful mathematical artwork. I thoroughly enjoyed myself!

I knew that fellow mathematical puzzle maker, Simona Prilogan,  was from Romania, but I didn’t realize until after I returned home that this castle is in her hometown! She recently wrote about the castle and her experience growing up in Hunedoara. As a child, she was chosen to write a poem about Romania’s president and present the poem at this very castle.

Now I’ll tell you a little something about the number 1164:

  • 1164 is a composite number.
  • Prime factorization: 1164 = 2 × 2 × 3 × 97, which can be written 1164 = 2² × 3 × 97
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1164 has exactly 12 factors.
  • Factors of 1164: 1, 2, 3, 4, 6, 12, 97, 194, 291, 388, 582, 1164
  • Factor pairs: 1164 = 1 × 1164, 2 × 582, 3 × 388, 4 × 291, 6 × 194, or 12 × 97
  • Taking the factor pair with the largest square number factor, we get √1164 = (√4)(√291) = 2√291 ≈ 34.117444

1164 is the sum of consecutive prime numbers FOUR different ways:
It’s the sum of the eighteen prime numbers from 29 to 103.
97 + 101 + 103 + 107 + 109 + 113 + 127 + 131+ 137 + 139 = 1164; that’s ten consecutive primes,
281 + 283 + 293 + 307 = 1164; that’s four consecutive primes, and
577 + 587 = 1164; that’s two consecutive primes.

1164 is the hypotenuse of a Pythagorean triple:
780-864-1164 which is 12 times (65-72-97)

1164 is palindrome 969 in BASE 11 because 9(11²) + 6(11) + 9(1) = 1164,
and it’s 345 in BASE 19 because 3(19²) + 4(19) + 5(1) = 1164

 

1162 Is a Pentagonal Number

1162 is the 28th pentagonal number because (3(28²) – 28)/2 = 1162. I created the graphic below so you can see all the pentagonal numbers up to 1162 and figure out why all of them are called pentagonal numbers.

Here’s more about the number 1162:

  • 1162 is a composite number.
  • Prime factorization: 1162 = 2 × 7 × 83
  • 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 1162 has exactly 8 factors.
  • Factors of 1162: 1, 2, 7, 14, 83, 166, 581, 1162
  • Factor pairs: 1162 = 1 × 1162, 2 × 581, 7 × 166, or 14 × 83
  • 1162 has no square factors that allow its square root to be simplified. √1162 ≈ 34.08812

1162 is a palindrome in three bases:
It’s 343 in BASE 19 because 3(19²) + 4(19) + 3(1) = 1162,
2I2 in BASE 20 (I is 18 base 10) because 2(20²) + 18(20) + 2(1) = 1162, and
1G1 in BASE 27 (G is 16 base 10) because 27² + 16(27) + 1 = 1162

What Kind of Shape is 1156 In?

One of the shapes that 1156 is in is a pentagon. Below you see 1156 tiny squares arranged into a pentagon.

If you study the graphic above you will also see that 1156 is 1 more than five times the 21st triangular number.

The first few centered pentagonal numbers are 1, 6, 16, 31, 51, 76, 106. Double any centered pentagonal number 106 or greater, and you will get a number that can be represented as 552 in some base even if it’s a base nobody uses. Let’s see it at work here:
1156 × 2 = 2312, and 2312 is 552 in BASE 21

Why is that true? Because of the formula that calculates centered pentagonal numbers. For example, we know 1156 is the 22nd centered pentagonal number from
(5(21²) + 5(21) + 2(1))/2 = 1156

Written another way it looks like 1 more than 5 times the 21st triangular number:
5(21)(22)/2 + 1 = 1156

1156 is also a perfect square!

Not only is it a perfect square but it is the square of a composite number. The following two squares show how it can be made with multiple squares of its prime factors.

  • 1156 is a composite number.
  • Prime factorization: 1156 = 2 × 2 × 17 × 17, which can be written 1156 = 2²× 17²
  • The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 × 3 = 9. Therefore 1156 has exactly 9 factors.
  • Factors of 1156: 1, 2, 4, 17, 34, 68, 289, 578, 1156
  • Factor pairs: 1156 = 1 × 1156, 2 × 578, 4 × 289, 17 × 68, or 34 × 34
  • 1156 is a perfect square. √1156 = 34

Another shape where you can find 1156 are these two Pythagorean triple triangles:
644-960-1156 and 544-1020-1156

1156 is also the sum of the twenty-four prime numbers from 5 to 101.

1156 is palindrome 1120211 in BASE 3 because 3⁶+3⁵+2(3⁴)+2(3²)+3+1=1156

Square number 1156 also looks like some other square numbers when it is written in these other bases:
It’s 961 in BASE 11 because 9(11²) + 6(11) + 1(1) = 1156,
484 in BASE 16 because 4(16²) + 8(16) + 4(1) = 1156,
400 in BASE 17 because 4(17²) = 1156,
169 in BASE 31 because 31² + 6(31) + 9(1) = 1156,
144 in BASE 32 because 32² + 4(32) + 4(1) = 1156,
121 in BASE 33 because 33² + 2(33) + 1(1) = 1156,
and 100 in BASE 34 because 34² = 1156

 

1141 is a Centered Hexagonal Number

20³ – 19³ = 1141 so 1141 is the 20th centered hexagonal number:

Here’s more about the number 1141:

  • 1141 is a composite number.
  • Prime factorization: 1141 = 5 × 163
  • 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 1141 has exactly 4 factors.
  • Factors of 1141: 1, 5, 163, 1141
  • Factor pairs: 1141 = 1 × 1141 or 5 × 163
  • 1141 has no square factors that allow its square root to be simplified. √1141 ≈ 33.77869

5º(31²) + 5¹(31¹) + 5²(31º) = 1141

1141 is palindrome 181 in BASE 30 because 30² + 8(30) + 1 = 1141

1140 is the 18th Tetrahedral Number

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

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