1100 Now THIS Is a Horse Race!

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!

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1099 and Level 5

The allowable common factors fro 8 and 24 are 2, 4, and 8. Which one of those should you choose? Find a different place to start the puzzle and you shouldn’t have to guess and check to see if you were right.

Print the puzzles or type the solution in this excel file: 12 factors 1095-1101

Here are some facts about the number 1099:

  • 1099 is a composite number.
  • Prime factorization: 1099 = 7 × 157
  • 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 1099 has exactly 4 factors.
  • Factors of 1099: 1, 7, 157, 1099
  • Factor pairs: 1099 = 1 × 1099 or 7 × 157
  • 1099 has no square factors that allow its square root to be simplified. √1099 ≈ 33.15117

1099 = 1 + 0 + 999 + 99. Thank you Stetson.edu for that fun fact.

1099 is the sum of the 13 prime numbers from 59 to 109.
That’s a fact that would take little effort to memorize!

1099 is also the sum of these prime numbers:
139 + 149 +  151 + 157 + 163 + 167 + 173 = 1099
359 + 367 + 373 = 1099

1099 is the hypotenuse of a Pythagorean triple:
595-924-1099 which is 7 times (85-132-157)

1099 is repdigit 777 in BASE 12 because 7(12² + 12 + 1) = 7(157) = 1099
1099 is palindrome 4D4 in BASE 15 (D is 13 in base 10)
because 4(15²) + 13(15) + 4(1) = 1099

1098 a Lucky Level 4 Puzzle?

You are lucky that this puzzle has some easy clues in it. You will have no problem getting started. Be warned, later on, you may not feel so lucky! I’m sure you can solve it if you keep with it.

Print the puzzles or type the solution in this excel file: 12 factors 1095-1101

Now, here are a few facts about the number 1098:

  • 1098 is a composite number.
  • Prime factorization: 1098 = 2 × 3 × 3 × 61, which can be written 1098 = 2 × 3² × 61
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1098 has exactly 12 factors.
  • Factors of 1098: 1, 2, 3, 6, 9, 18, 61, 122, 183, 366, 549, 1098
  • Factor pairs: 1098 = 1 × 1098, 2 × 549, 3 × 366, 6 × 183, 9 × 122, or 18 × 61,
  • Taking the factor pair with the largest square number factor, we get √1098 = (√9)(√122) = 3√122 ≈ 33.13608

This first fact from Stetson.edu uses only digits found in 1098. It makes 1098 look pretty lucky:
1098 = 11 + 0 + 999 + 88

1098 is the sum of four consecutive prime numbers:
269 + 271 + 277 + 281 = 1098

1098 is the hypotenuse of a Pythagorean triple:
198-1080-1098 which is 18 times (11-60-61)

1098 is a palindrome when it is written in three different bases:
It’s 2112 in BASE 8 because 2(8³) + 8² + 8 + 2(1) = 1098,
909 in BASE 11 because 9(11² + 1) = 9(122) = 1098, and it’s
666 in BASE 13. Oh my! How unlucky can you get? Why does it have two unlucky numbers, 666 and 13? Because 6(13² + 13 + 1) = 6(183) = 1098.

 

1097 and Level 3

72 and 27 are mirror images of each other. What is the largest number that will divide evenly into both of them? Put the answer to that question under the x, and you will have completed the first step in solving this multiplication table puzzle.

Print the puzzles or type the solution in this excel file: 12 factors 1095-1101

Here’s a little bit more about the number 1097:

  • 1097 is a prime number.
  • Prime factorization: 1097 is prime.
  • The exponent of prime number 1097 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1097 has exactly 2 factors.
  • Factors of 1097: 1, 1097
  • Factor pairs: 1097 = 1 × 1097
  • 1097 has no square factors that allow its square root to be simplified. √1097 ≈ 33.12099

How do we know that 1097 is a prime number? If 1097 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1097 ≈ 33.1. Since 1097 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1097 is a prime number.

1097 is the final prime number in the prime triplet, 1091-1093-1097.

1097 is the sum of two squares:
29² + 16² = 1097

1097 is the hypotenuse of a primitive Pythagorean triple:
585-928-1097 calculated from 29² – 16², 2(29)(16), 29² + 16²

Here’s another way we know that 1097 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 29² + 16² = 1097 with 29 and 16 having no common prime factors, 1097 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1097 ≈ 33.1. Since 1097 is not divisible by 5, 13, 17, or 29, we know that 1097 is a prime number.

 

 

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 puzzle won’t be in the usual places. Can you figure out where they need to go?

Print the puzzles or type the solution in this excel file: 12 factors 1095-1101

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)

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.)

Print the puzzles or type the solution in this excel file: 12 factors 1095-1101

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