1096 and Level 2

There are 17 clues in this level 2 puzzle. Two of those clues are 60 and three of them are 8. In a regular 12 × 12 multiplication table, both of those numbers appear 4 times each. The factors for this multiplication table won’t be in the usual places. Can you figure out where they need to go?

Here is a little bit about the number 1096:

  • 1096 is a composite number.
  • Prime factorization: 1096 = 2 × 2 × 2 × 137, which can be written 1096 = 2³ × 137
  • The exponents in the prime factorization are 1 and 3. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1096 has exactly 8 factors.
  • Factors of 1096: 1, 2, 4, 8, 137, 274, 548, 1096
  • Factor pairs: 1096 = 1 × 1096, 2 × 548, 4 × 274, or 8 × 137
  • Taking the factor pair with the largest square number factor, we get √1096 = (√4)(√274) = 2√274 ≈ 33.10589

1096 is the hypotenuse of one Pythagorean triple:
704-840-1096 which is 8 times (88-105-137)

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1095 and Level 1

This awkward-looking puzzle is as simple as clockwork to solve. Put a 5 above the 55 in the top row and another 5 before the 30 in the first column. Then write what number the big hand is pointing to when it is 55 minutes after the hour and so forth until you have written the numbers from 1 to 12 in both the first column and the top row. (You will have to figure out what numbers go with the column and row without clues.)

That puzzle had something to do with a clock, while the number 1095 has something to do with the number of days in three non-leap years. Here are some other facts about it:

1 + 0 + 9 + 5 = 15, so 1095 can be evenly divided by 3. Since its last digit is 5, it is also divisible by 5.

  • 1095 is a composite number.
  • Prime factorization: 1095 = 3 × 5 × 73
  • 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 1095 has exactly 8 factors.
  • Factors of 1095: 1, 3, 5, 15, 73, 219, 365, 1095
  • Factor pairs: 1095 = 1 × 1095, 3 × 365, 5 × 219, or 15 × 73
  • 1095 has no square factors that allow its square root to be simplified. √1095 ≈ 33.09078

1095 is the hypotenuse of FOUR Pythagorean triples:
81-1092-1095 which is 3 times (27-364-365)
228-1071-1095 which is 3 times (76-357-365)
657-876-1095 which is (3-4-5) times 219
720-825-1095 which is 15 times (48-55-73)

1094 and Level 6

Should you choose 4 or 8 as the common factor of 32 and 16 in this puzzle?
Is 3 or 9 the common factor needed for 9 and 18?
And is 4 or 6 the common factor for 36 and 12 that will make this puzzle work?
In each of those cases, only one of those factors will work. Which one will it be?

The other clues will help you know where to logically start this puzzle. There is no need to guess and check. The entire puzzle can be solved using logic. Have fun!

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

Now I’ll tell you something about the number 1094:

  • 1094 is a composite number.
  • Prime factorization: 1094 = 2 × 547
  • 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 1094 has exactly 4 factors.
  • Factors of 1094: 1, 2, 547, 1094
  • Factor pairs: 1094 = 1 × 1094 or 2 × 547
  • 1094 has no square factors that allow its square root to be simplified. √1094 ≈ 33.07567

1094 is palindrome 2A2 in BASE 21 (A is 10 base 10) because 2(21²) + 10(21) + 2(1) = 1094

 

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!

 

1092 Predictable Factor Trees

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

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

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

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

1088 and Level 2

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

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