Author: ivasallay

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!

 

1092 Predictable Factor Trees

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A couple of years ago on Memorial Day weekend, we bought two peach trees from a local nursery. Those trees have grown bigger, and we will get some peaches this year. The smaller tree has lots of fruit growing on it and will need some attention because the branches will be too small to support the weight of all that fruit. The bigger tree has exactly one peach growing on it.

These factor trees for the number 1092 are a lot more predictable than those peach trees: No matter which of its factor pairs you use, you will always eventually get
2² × 3 × 7 × 13 = 1092.

Here are some more facts about 1092:

  • 1092 is a composite number.
  • Prime factorization: 1092 = 2 × 2 × 3 × 7 × 13, which can be written 1092 = 2² × 3 × 7 × 13
  • 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 1092 has exactly 24 factors.
  • Factors of 1092: 1, 2, 3, 4, 6, 7, 12, 13, 14, 21, 26, 28, 39, 42, 52, 78, 84, 91, 156, 182, 273, 364, 546, 1092
  • Factor pairs: 1092 = 1 × 1092, 2 × 546, 3 × 364, 4 × 273, 6 × 182, 7 × 156, 12 × 91, 13 × 84, 14 × 78, 21 × 52, 26 × 42, or 28 × 39
  • Taking the factor pair with the largest square number factor, we get √1092 = (√4)(√273) = 2√273 ≈ 33.04542

(12 × 13 × 14)/2 = 1092
Even though 1092 can claim that cool fact, it will actually make the next number a STAR!

1092 is the sum of the twelve prime numbers from 67 to 113:
67 + 71 + 73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 = 1092

1092 has a lot of factors, but it is the hypotenuse of only one Pythagorean triple:
420-1008-1092 which is (5-12-13) times 84

Look how cool 1092 is in some other bases:

Not only does it use only 0’s and 1’s in base 2 but also in consecutive bases 3 and 4:
It’s 10001000100 in BASE 2 because 2¹º + 2⁶ + 2² = 1092,
1111110 in BASE 3 because 3⁶ + 3⁵ + 3⁴ + 3³ + 3² + 3¹ = 1092,
and 101010 in BASE 4 because  4⁵ + 4³ + 4¹ = 1092

I like the way it looks in consecutive bases 12 and 13:
It’s 770 in BASE 12, because 7(12² + 12) = 7(156) = 1092
and 660 in BASE 13 because 6(13² + 13) = 6(182) = 1092

And its repdigit 444 in BASE 16 because 4(16² + 16 + 1) = 1092

Some of these facts about 1092 were predictable and some were not, but I have enjoyed learning all of them and hope that you have too.

1091 and Level 5

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Can you figure out where to put all the numbers from 1 to 10 in both the first column and the top row so that those factors and the clues can become a multiplication table? Some of the clues might be a little tricky, but I’m sure you can figure them all out.

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

Here are a few facts about the number 1091:

  • 1091 is a prime number.
  • Prime factorization: 1091 is prime.
  • The exponent of prime number 1091 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1091 has exactly 2 factors.
  • Factors of 1091: 1, 1091
  • Factor pairs: 1091 = 1 × 1091
  • 1091 has no square factors that allow its square root to be simplified. √1091 ≈ 33.03029

How do we know that 1091 is a prime number? If 1091 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1091 ≈ 33. Since 1091 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1091 is a prime number.

1091 is the first prime number in the prime triplet (1091, 1093, 1097). It is also the middle number in the prime triplet (1087, 1091, 1093).

1091 looks interesting when it is written in some other bases:
It’s 13331 in BASE 5 because 1(5⁴) + 3(5³) + 3(5²) + 3(5) + 1(1) = 1091,
3D3 in BASE 17 (D is 13 base 10) because 3(17²) + 13(17) + 3(1) = 1091,
and it’s 123 in BASE 32 because 1(32²) + 2(32) + 3(1) = 1091

1090 and Level 4

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This puzzle has both 54 and 56 as clues. Many people get the factors involved (6, 7, 8, 9) mixed up. Remember these two multiplication facts:
6 7 × 8 9 = 54
6 7 × 8 9 = 56
The closer factors make the bigger number.

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

Now I’ll share some information about the number 1090:

  • 1090 is a composite number.
  • Prime factorization: 1090 = 2 × 5 × 109
  • 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 1090 has exactly 8 factors.
  • Factors of 1090: 1, 2, 5, 10, 109, 218, 545, 1090
  • Factor pairs: 1090 = 1 × 1090, 2 × 545, 5 × 218, or 10 × 109
  • 1090 has no square factors that allow its square root to be simplified. √1090 ≈ 33.01515

1090 is the sum of the fourteen prime numbers from 47 to 107.

1090 is the sum of two squares two different ways:
27² + 19² = 1090
33² +  1² = 1090

1090 is the hypotenuse of four Pythagorean triples:
66-1088-1090 calculated from 2(33)(1), 33² –  1², 33² +  1²;
it is also 2 times (33-544-545),
368-1026-1090 calculated from 27² – 19² , 2(27)(19) , 27² + 19²;
it is also 2 times (184-513-545),
600-910-1090 which is 10 times (60-91-109)
654-872-1090 which is (3-4-5) times 218

1090 is a palindrome in two different bases:
It’s 1441 in BASE 9 because 1(9³) + 4(9²) + 4(9) + 1(1) = 1090
101 in BASE 33 because 33² + 1 = 1090

1089 Perfect Squares

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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 n 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

OEIS.org 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

1088 and Level 2

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This puzzle has three rows with three numbers in each and three columns with three numbers in each. Find the biggest number that is 10 or less that is a common factor of each set of three numbers, and you will be well on your way of solving the entire puzzle. Can you do it?

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

Now I’ll share some information about the number 1088:

  • 1088 is a composite number.
  • Prime factorization: 1088 = 2 × 2 × 2 × 2 × 2 × 2 × 17, which can be written 1088 = 2⁶ × 17
  • The exponents in the prime factorization are 6, and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1) = 7 × 2 = 14. Therefore 1088 has exactly 14 factors.
  • Factors of 1088: 1, 2, 4, 8, 16, 17, 32, 34, 64, 68, 136, 272, 544, 1088
  • Factor pairs: 1088 = 1 × 1088, 2 × 544, 4 × 272, 8 × 136, 16 × 68, 17 × 64, or 32 × 34
  • Taking the factor pair with the largest square number factor, we get √1088 = (√64)(√17) = 8√17 ≈ 32.98485

Since 1088 = 32 × 34, we know the next number will be a square number.

1088 is the hypotenuse of a Pythagorean triple:
512-960-1088 which is (8-15-17) times 64

1088 is the sum of two consecutive prime numbers:
541 +547 = 1088

1088 looks interesting when written in some other bases:
It’s 3113 in BASE 7 because 3(7³) + 1(7²) + 1(7) +3(1) = 1088,
WW in BASE 33 (W is 32 base 10) Because 32(33) + 32(1) = 32(33 + 1) = 1088,
and it’s W0 in BASE 34 because 32(34) = 1088

1087 and Level 3

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Using logic, start with the clue on the top row and work yourself down row by row filling in the appropriate factors while you go. You might find this level 3 puzzle a little tricky near the bottom of the puzzle, so I didn’t want to wait to share it with you. Happy factoring!

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

1087 is the first prime since 1069, which was 18 numbers ago! What else can I tell you about it?

  • 1087 is a prime number.
  • Prime factorization: 1087 is prime.
  • The exponent of prime number 1087 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1087 has exactly 2 factors.
  • Factors of 1087: 1, 1087
  • Factor pairs: 1087 = 1 × 1087
  • 1087 has no square factors that allow its square root to be simplified. √1087 ≈ 32.96968

How do we know that 1087 is a prime number? If 1087 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1087 ≈ 33. Since 1087 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1087 is a prime number.

1087 is also palindrome 767 in BASE 12 because
7(12²) + 6(12) + 7(1) = 1087

1086 and Level 6

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The Find the Factors 1-12 puzzles allow 36 and 72 to use three common factors: 6, 9, and 12. Which one should you choose?  Likewise, 16 and 4’s permitted common factors are 2 and 4.

Sure you could guess and check to solve this puzzle, but that might frustrate you. I assure you that you can solve this puzzle by using logic.

If you get stuck getting started, here are a couple of things to consider:
Can both 72’s in the puzzle be 8×9? How about 6×12?
(36 = 3 × 12), (30 = 3 × 10), and (33 = 3 × 11) Which two of those clues MUST use both of the 3’s?

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

Now I’ll share a few facts about the number 1086:

  • 1086 is a composite number.
  • Prime factorization: 1086 = 2 × 3 × 181
  • 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 1086 has exactly 8 factors.
  • Factors of 1086: 1, 2, 3, 6, 181, 362, 543, 1086
  • Factor pairs: 1086 = 1 × 1086, 2 × 543, 3 × 362, or 6 × 181
  • 1086 has no square factors that allow its square root to be simplified. √1086 ≈ 32.95451

1086 is the hypotenuse of a Pythagorean triple:
114-1080-1086 which is 6 times (19-180-181)

1086 is a palindrome in three other bases:
It’s 8A8 in BASE 11 (A is 10 base 10) because 8(121) + 10(11) + 8(1) = 1086,
303 in BASE 19 because 3(19² + 1) = 1086, and
141 in BASE 31 because 1(31²) + 4(31) + 1(1) = 1086

1085 and Level 5

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Here’s another tricky level 5 puzzle for you to solve. Use logic, not guess and check, and you’ll do great!

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

Here are a few facts about the number 1085:

  • 1085 is a composite number.
  • Prime factorization: 1085 = 5 × 7 × 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 1085 has exactly 8 factors.
  • Factors of 1085: 1, 5, 7, 31, 35, 155, 217, 1085
  • Factor pairs: 1085 = 1 × 1085, 5 × 217, 7 × 155, or 31 × 35
  • 1085 has no square factors that allow its square root to be simplified. √1085 ≈ 32.93934

31 × 35 = 1085 means we are  only 4 numbers away from the next perfect square and
33² – 2² = 1085

1085 is the hypotenuse of a Pythagorean triple:
651-868-1085 which is (3-4-5) times 217

1085 looks interesting when it is written using a different base:
It’s 5005 in BASE 6 because 5(6³ + 1) = 1085,
765 in BASE 12 because 7(144) + 6(12) + 5(1) = 1085,
656 in BASE 13 because 6(13²) + 5(13) + 6(1) = 1085
VV in BASE 34 (V is 31 base 10) because 31(34) + 31(1) = 1085, and
it’s V0 in BASE 35 because 31(35) = 1085

1084 and Level 4

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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