1093 is a STAR!

(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 Stetson.edu 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!

 

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1089 Perfect Squares

The nine clues in today’s puzzle are all perfect squares. They are all you need to find all the factors that can turn this puzzle into a multiplication table . . . but with the rows and columns not in the typical order:

Print the puzzles or type the solution in this excel file: 10-factors-1087-1094

1089 is also a perfect square.

The previous perfect square was 32² = (33 – 1)² = 33² + 1 – 2(33) = 1024
The next perfect square will be 34² = (33 + 1)² = 33² + 1 + 2(33) = 1156

Here’s a little more about the number 1089:

  • 1089 is a composite number.
  • Prime factorization: 1089 = 3 × 3 × 11 × 11, which can be written 1089 = 3²× 11²
  • 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 1089 has exactly 9 factors.
  • Factors of 1089: 1, 3, 9, 11, 33, 99, 121, 363, 1089
  • Factor pairs: 1089 = 1 × 1089, 3 × 363, 9 × 121, 11 × 99, or 33 × 33
  • 1089 is a perfect square. √1089 = 33

1, 9, 121, and 1089 are all perfect square factors of 1089.

1089 can be 3 × 3 perfect squares arranged on an 11 × 11 perfect square grid:

1089 can also be 11 × 11 perfect squares arranged on a 3 × 3 perfect square grid:

Not only is 1089 the 33rd perfect square, but it is also the sum of the first 33 odd numbers. Note that the nth perfect square is also the sum of the first odd numbers:

I’m not attempting to make a picture of this nine-sided shape, but 1089 is the 18th nonagonal number because 18(7(18) – 5)/ 2 = 1089,
or written another way 7(18²)/2 – 5(18)/2 = 1089.

1089 is the sum of five consecutive prime numbers:
199 + 211 + 223 + 227 + 229 = 1089

Stetson.edu informs us that 9 × 1089 = 9801

1089 looks rather square when it is written in several other bases:
It’s 900 in BASE 11 because 9(11²) = 1089,
441 in BASE 16 because  4(16²) + 4(16) + 1(1) = 1089,
169 in BASE 30 because 1(30²) + 6(30) + 9(1) = 1089,
144 in BASE 31 because 1(31²) + 4(31) + 4(1) = 1089,
121 in BASE 32 because 1(32²) + 2(32) + 1(1) = 1089,
100 in BASE 33 because 1(33²) = 1089

1084 and Level 4

Where should you put all the numbers 1 to 12 in both the top row and the first column?  You will have to think about it and use logic. Some of the clues might be tricky, but you’ll figure it all out.

Print the puzzles or type the solution in this excel file: 12 factors 1080-1086

Here are some facts about the number 1084:

  • 1084 is a composite number.
  • Prime factorization: 1084 = 2 × 2 × 271, which can be written 1084 = 2² × 271
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 1084 has exactly 6 factors.
  • Factors of 1084: 1, 2, 4, 271, 542, 1084
  • Factor pairs: 1084 = 1 × 1084, 2 × 542, or 4 × 271
  • Taking the factor pair with the largest square number factor, we get √1084 = (√4)(√271) = 2√271 ≈ 32.92416

1084 is the sum of the 18 prime numbers from 23 to 101.

It is also the sum of six consecutive prime numbers:
167 + 173 + 179 + 181 + 191 + 193  = 1084

STOP! Look How Cool a Number 1080 Is!

What can I tell you about the number 1080?  Lots of things!

The sum of the interior angles of an eight-sided polygon such as a stop sign is 1080°.

The sum of the interior angles of an octagon is 6(180°) = 1080°. Convex or Concave, it doesn’t matter, the sum of those interior angles of an eight-sided polygon will still be 1080°, as illustrated below:

There is only one number less than 1080 that has as many factors as 1080 does. What was that number? 840. How many factors does1080 have? 32. Wow!

  • 1080 is a composite number.
  • Prime factorization: 1080 = 2 × 2 × 2 × 3 × 3 × 3 × 5, which can be written 1080 = 2³ × 3³ × 5.
  • The exponents in the prime factorization are 3, 3, and 1. Adding one to each and multiplying we get (3 + 1)(3 + 1)(1 + 1) ) = 4 × 4 × 2 = 32. Therefore 1080 has exactly 32 factors.
  • Factors of 1080: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 27, 30, 36, 40, 45, 54, 60, 72, 90, 108, 120, 135, 180, 216, 270, 360, 540, 1080
  • Factor pairs: 1080 = 1 × 1080, 2 × 540, 3 × 360, 4 × 270, 5 × 216, 6 × 180, 8 × 135, 9 × 120, 10 × 108, 12 × 90, 15 × 72, 18 × 60, 20 × 54, 24 × 45, 27 × 40 or 30 × 36
  • Taking the factor pair with the largest square number factor, we get √1080 = (√36)(√30) = 6√30 ≈ 32.86335

1080 has 28 composite factors and is the smallest number that can make that claim. (Of its 32 factors, all are composite numbers except 1, 2, 3, and 5). That’s more than 840’s 27 composite factors. (Its 32 factors minus 1, 2, 3, 5, and 7)

Here are a couple of the MANY possible factor trees for 1080:

1080 is the sum of these four consecutive prime numbers:
263 + 269 + 271 + 277 = 1080

1080 is the sum of four consecutive powers of three:
3⁶ + 3⁵ + 3⁴ + 3³ = 1080

1080 is the hypotenuse of a Pythagorean triple:
648-864-1080 which is (3-4-5) times 216

Note that 5(6³) = 5(216) = 1080 so 1080 is 500 in BASE 6.
It’s palindrome 252 in BASE 22 because 2(22²) + 5(22) + 2(1) = 1080,
UU in BASE 35 (U is 30 base 10) because 30(35) + 30(1) = 30(36) = 1080,
and it’s U0 in BASE 36 because 30(36) = 1080

And now I’ll STOP writing about how cool 1080 is.

1079 An Easier Find The Factors Challenge?

This find the factors challenge puzzle might be easier than most of the challenge puzzles are, but it will still give you plenty of reasons to think about what factors you should put where. You need to put all the numbers from 1 to 10 in each of the four bold areas so that those numbers are the factors of the given clues. There is only one solution. Can you find it?

Print the puzzles or type the solution in this excel file: 10-factors-1073-1079

Here’s a little about the number 1079:

  • 1079 is a composite number.
  • Prime factorization: 1079 = 13 × 83
  • 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 1079 has exactly 4 factors.
  • Factors of 1079: 1, 13, 83, 1079
  • Factor pairs: 1079 = 1 × 1079 or 13 × 83
  • 1079 has no square factors that allow its square root to be simplified. √1079 ≈ 32.84814

1079 is the sum of three consecutive prime numbers:
353 + 359 + 367 = 1079

1079 is also the hypotenuse of a Pythagorean triple:
415-996-1079 which is (5-12-13) times 83

1056 How to Tile a 32 × 33 Floor

32 × 33 = 1056, and Stetson.edu informs us that those are the smallest rectangular dimensions that can be tiled with different perfect squares.

It isn’t difficult to do the tiling. All you have to remember is 32 × 33 and to put an 18 × 18 tile in a corner. The rest of the perfect square tiles seem to almost fall into place as this gif I made illustrates:

Tiling a 32 × 33 Rectangle

make science GIFs like this at MakeaGif
Now I’ll tell you a little about the number 1056:
  • 1056 is a composite number.
  • Prime factorization: 1056 = 2 × 2 × 2 × 2 × 2 × 3 × 7, which can be written 1056 = 2⁵ × 3 × 7
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 × 2 × 2 = 24. Therefore 1056 has exactly 24 factors.
  • Factors of 1056: 1, 2, 3, 4, 6, 8, 11, 12, 16, 22, 24, 32, 33, 44, 48, 66, 88, 96, 132, 176, 264, 352, 528, 1056
  • Factor pairs: 1056 = 1 × 1056, 2 × 528, 3 × 352, 4 × 264, 6 × 176, 8 × 132, 11 × 96, 12 × 88, 16 × 66, 22 × 48, 24 × 44, or 32 × 33
  • Taking the factor pair with the largest square number factor, we get √1056 = (√16)(√66) = 4√66 ≈ 32.49615
Since 1056 is the product of consecutive numbers, 32 × 33, it is the sum of the first 32 even numbers:
2 + 4 + 6 + 8 + 10 + . . .  +56 + 58 + 60 + 62 + 64 = 1056

 

How Can You Count These 1054 Tiny Squares?

There are 1054 tiny squares in the image below, making 1054 a centered triangular number. How can you know that I’m not pulling the wool over your eyes about the number of tiny squares? Here are a few ways that you can quickly count all of them.

If you start with the yellow square in the center and count outward each succeeding triangle you will get 1 yellow square + 3 green squares + 6 blue squares + 9 purple squares + 12 red squares + 15 orange squares, etc. until you reach the final 78 blue squares:
1 + 3 + 6 + 9 + 12 + 15 + . . . + 78
= 1 + 3(1 + 2 + 3 + 4 + 5 + . . . + 26)
= 1 + 3(26*27)/2 = 1 + 3(351) = 1054

Using a little bit of algebra, you can show that
1 + 3(26*27)/2 = (3(26²) + 3(26) + 2)/2 = 1054

You can divide the centered triangle above into three triangles as I also did in the graphic. The three triangles represent the 25th, the 26th, and the 27th triangular numbers. Adding them up you get:
25(26)/2 + 26(27)/2 + 27(28)/2 = 325 + 351 + 378 = 1054

Here is some more information about the number 1054:

  • 1054 is a composite number.
  • Prime factorization: 1054 = 2 × 17 × 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 1054 has exactly 8 factors.
  • Factors of 1054: 1, 2, 17, 31, 34, 62, 527, 1054
  • Factor pairs: 1054 = 1 × 1054, 2 × 527, 17 × 62, or 31 × 34
  • 1054 has no square factors that allow its square root to be simplified. √1054 ≈ 32.465366

1054 is the sum of six consecutive prime numbers:
163 + 167 + 173 + 179 + 181 + 191 = 1054

1054 is the hypotenuse of a Pythagorean triple:
496-930-1054 which is (8-15-17) times 62

1054 looks interesting when it is written in some other bases:
It’s 4A4 in BASE 15 (A is 10 base 10) because 4(15²) + 10(15) + 4(1) = 1054
1C1 in BASE 27 (C is 12 base 10) because 27² + 12(27) + 1 = 1054
VV in BASE 33 (V is 31 base 10) because 31(33) + 31(1) = 31(34) = 1054
V0 in BASE 34 because 31(34) = 1054

 

 

 

 

1051 is the 21st Centered Pentagonal Number

1051 is the 21st centered pentagonal number. It is exactly 100 more than the previous centered pentagonal number because there are exactly 100 little blue squares on the outside-most pentagon in the graphic below.

Can you see the five triangles surrounding the center square? Each of them has the same number of tiny squares and indicates that 1051 is 1 more than five times the 20th triangular number:
1 + 5(20)(21)/2 = 1 + 50(21) = 1051

1049 and 1051 are twin primes.

  • 1051 is a prime number.
  • Prime factorization: 1051 is prime.
  • The exponent of prime number 1051 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1051 has exactly 2 factors.
  • Factors of 1051: 1, 1051
  • Factor pairs: 1051 = 1 × 1051
  • 1051 has no square factors that allow its square root to be simplified. √1051 ≈ 32.41913

How do we know that 1051 is a prime number? If 1051 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1051 ≈ 32.4. Since 1051 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1051 is a prime number.

1051 is a palindrome when it is written in three other bases:
It’s 737 in BASE 12 because 7(144) + 3(12) + 7(1) = 1051,
1H1 in BASE 25 (H is 17 base 10) because 25² +17(25) + 1 = 1051, and
151 in BASE 30 because 30² + 5(30) + 1 = 1051

1041 and Level 4

Any level 3 puzzle can be easily made into a level 4 puzzle by removing some restrictions on the order of the clues. If you can solve a level 3 puzzle, then this level 4 puzzle will be only a little more difficult to solve.

Print the puzzles or type the solution in this excel file: 10-factors-1035-1043

What have I found out about the number 1041?

  • 1041 is a composite number.
  • Prime factorization: 1041 = 3 × 347
  • 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 1041 has exactly 4 factors.
  • Factors of 1041: 1, 3, 347, 1041
  • Factor pairs: 1041 = 1 × 1041 or 3 × 347
  • 1041 has no square factors that allow its square root to be simplified. √1041 ≈ 32.26453

1041 is also a palindrome in three other bases:
It’s 13131 in BASE 5 because 5⁴ + 3(5³) + 5² +3(5) + 1 = 1041,
545 in BASE 14 because 5(14²) + 4(14) + 5(1) = 1041, and
1E1 in BASE 26 (E is 14 base 10) because 26² + 14(26) + 1 = 1041

1035 is the 23rd Hexagonal Number

1035 is the 23rd hexagonal number because of the way that it can be factored:
2(23²) – 23 = 1035,
(2(23) – 1)23 = 1035
or simply 45(23) = 1035.

Starting at the bottom of the hexagonal we see 1 yellow square, 5 green squares, 9 blue squares, 13 purple squares, 17 red squares, and 21 orange squares.

1, 5, 9, 13, 17, 21, . . . is an arithmetic progression or arithmetic sequence. The common difference between the numbers is 4.

The nth hexagonal number is the sum of the first n numbers in that arithmetic progression.
The first few hexagonal numbers form an arithmetic series: 1, 6, 15, 28, 45, 66 and so forth.
1035 is the sum of the first 23 numbers in the progression so it is the 23rd term in the series and the 23rd hexagonal number.

All hexagonal numbers are also triangular numbers. 1035 is the 45th triangular number because 45(46)/2 = 1035.

Starting in the lower left-hand corner of that triangle we see 1 yellow square, 2 green squares, 3 blue squares, 4 purple squares, 5 red squares, and 6 orange squares.

1, 2, 3, 4, 5, 6, . . .  is the simplest arithmetic progression there is. The common difference is 1.

The nth triangular number is the sum of the first n numbers in that arithmetic progression.

The first few triangular numbers form an arithmetic series: 1, 3,  6, 10, 15, 21, 28, 36, 45, 55, 66 and so forth.  (The blue triangular numbers are also hexagonal numbers.)
1035 is the sum of the first 45 numbers in the progression so it is the 45th term in the series and the 45th triangular number.

1035 is also the hypotenuse of one Pythagorean triple:
621-828-1035 which is (3-4-5) times 207

It is also a leg in several Pythagorean triples including
1035-1380-1725 which is (3-4-5) times 345

  • 1035 is a composite number.
  • Prime factorization: 1035 = 3 × 3 × 5 × 23, which can be written 1035 = 3² × 5 × 23
  • 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 = 23. Therefore 1035 has exactly 12 factors.
  • Factors of 1035: 1, 3, 5, 9, 15, 23, 45, 69, 115, 207, 345, 1035
  • Factor pairs: 1035 = 1 × 1035, 3 × 345, 5 × 207, 9 × 115, 15 × 69, or 23 × 45,
  • Taking the factor pair with the largest square number factor, we get √1035 = (√9)(√115) = 3√115 ≈ 32.1714