1110 Another Mystery

The common factors of 36 and 12 are 1, 2, 3, 4, 6, and 12. If this were a Find the Factors 1-12 puzzle, you would have to consider most of those common factors. Since it’s a 1 to 10 puzzle, most of those factors aren’t allowed. Can you figure out the solution to this mystery puzzle?

Print the puzzles or type the solution in this excel file: 10-factors-1102-1110

Here are a few facts about the number 1110:

  • 1110 is a composite number.
  • Prime factorization: 1110 = 2 × 3 × 5 × 37
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1110 has exactly 16 factors.
  • Factors of 1110: 1, 2, 3, 5, 6, 10, 15, 30, 37, 74, 111, 185, 222, 370, 555, 1110
  • Factor pairs: 1110 = 1 × 1110, 2 × 555, 3 × 370, 5 × 222, 6 × 185, 10 × 111, 15 × 74, or 30 × 37
  • 1110 has no square factors that allow its square root to be simplified. √1110 ≈ 33.31666

1110 is the hypotenuse of FOUR Pythagorean triples:
342-1056-1110 which is 6 times (57-176-185)
360-1050-1110 which is 30 times (12-35-37)
624-918-1110 which is 6 times (104-153-185)
666-888-1110 which is (3-4-5) times 222

1110 looks very cool in some other bases:
It’s 5050 in BASE 6 because 5(6³) + 5(6) = 5(222) = 1110,
456 in BASE 16 because 4(16²) + 5(16) + 6(1) = 1110, and
UU in BASE 36 (U is 30 in base 10) because 30(36) + 30(1) = 30(37) = 1110

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1109 Mystery Level

Is this puzzle easy to solve or difficult? That’s part of the mystery. I hope you will give it a try and figure it out.

Print the puzzles or type the solution in this excel file: 10-factors-1102-1110

Here is some information about the number 1109:

  • 1109 is a prime number.
  • Prime factorization: 1109 is prime.
  • The exponent of prime number 1109 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1109 has exactly 2 factors.
  • Factors of 1109: 1, 1109
  • Factor pairs: 1109 = 1 × 1109
  • 1109 has no square factors that allow its square root to be simplified. √1109 ≈ 33.30165

How do we know that 1109 is a prime number? If 1109 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1109 ≈ 33.3. Since 1109 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1109 is a prime number.

25² + 22² = 1109

1109 is the hypotenuse of a Pythagorean triple:
141-1100-1109 calculated from 25² – 22², 2(25)(22), 25² + 22²

Here’s another way we know that 1109 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 25² + 22² = 1109 with 25 and 22 having no common prime factors, 1109 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1109 ≈ 33.3. Since 1109 is not divisible by 5, 13, 17, or 29, we know that 1109 is a prime number.

1109 is palindrome 919 in BASE 11 because 9(121) + 1(11) + 9(1) = 1109

1108 and Level 6

This level 6 puzzle has eleven clues. Which ones give away the most logical place for you to start it? I hope you have a lot of fun solving this one!

Print the puzzles or type the solution in this excel file: 10-factors-1102-1110

Now I’ll share some facts about the number 1108:

  • 1108 is a composite number.
  • Prime factorization: 1108 = 2 × 2 × 277, which can be written 1108 = 2² × 277
  • 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 1108 has exactly 6 factors.
  • Factors of 1108: 1, 2, 4, 277, 554, 1108
  • Factor pairs: 1108 = 1 × 1108, 2 × 554, or 4 × 277
  • Taking the factor pair with the largest square number factor, we get √1108 = (√4)(√277) = 2√277 ≈ 33.28663

1108 is the hypotenuse of a Pythagorean triple:
460-1008-1108 which is 4 times (115-252-277)

1108 is a palindrome when it is written in three other bases:
It’s 454 in BASE 16 because 4(16²) + 5(16) + 4(1) = 1108,
3E3 in BASE 17 (E is 14 base 10) because 3(17²) +14(17) +3(1) = 1108, and
1E1 in BASE 27 because 27² + 14(27) + 1 = 1108

1107 and Level 5

Some of this puzzle might be a little tricky, but you won’t allow it to trick you, right? Of course not!

Print the puzzles or type the solution in this excel file: 10-factors-1102-1110

Let me tell you something about the number 1107:

  • 1107 is a composite number.
  • Prime factorization: 1107 = 3 × 3 × 3 × 41, which can be written 1107 = 3³ × 41
  • 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 1107 has exactly 8 factors.
  • Factors of 1107: 1, 3, 9, 27, 41, 123, 369, 1107
  • Factor pairs: 1107 = 1 × 1107, 3 × 369, 9 × 123, or 27 × 41
  • Taking the factor pair with the largest square number factor, we get √1107 = (√9)(√123) = 3√123 ≈ 33.27161

1107 is the hypotenuse of a Pythagorean triple:
243-1080-1107 which is 27 times (9-40-41)

1106 and Level 4

Can you use logic to figure out where all the numbers from 1 to 10 need to go in both the first column and the top row so that this puzzle can become a multiplication table? Give it a try. It’s fun!

Print the puzzles or type the solution in this excel file: 10-factors-1102-1110

Here is some information about the number 1106:

  • 1106 is a composite number.
  • Prime factorization: 1106 = 2 × 7 × 79
  • 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 1106 has exactly 8 factors.
  • Factors of 1106: 1, 2, 7, 14, 79, 158, 553, 1106
  • Factor pairs: 1106 = 1 × 1106, 2 × 553, 7 × 158, or 14 × 79
  • 1106 has no square factors that allow its square root to be simplified. √1106 ≈ 33.25658

1106 is repdigit 222 in BASE 23 because 2(23² + 23+ 1) = 2(553) = 1106

1104 and Level 3

If this were a Find the Factors 1-12 puzzle, the possible common factors for 12 and 48 would be 4, 6, and 12. But we can only have factors from 1 to 10 so only one of those common factors will work with this puzzle. If you know which one, you are well on your way to solving it.

Print the puzzles or type the solution in this excel file: 10-factors-1102-1110

Here are some facts about the number 1104:

  • 1104 is a composite number.
  • Prime factorization: 1104 = 2 × 2 × 2 × 2 × 3 × 23, which can be written 1104 = 2⁴ × 3 × 23
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1104 has exactly 20 factors.
  • Factors of 1104: 1, 2, 3, 4, 6, 8, 12, 16, 23, 24, 46, 48, 69, 92, 138, 184, 276, 368, 552, 1104
  • Factor pairs: 1104 = 1 × 1104, 2 × 552, 3 × 368, 4 × 276, 6 × 184, 8 × 138, 12 × 92, 16 × 69, 23 × 48, or 24 × 46
  • Taking the factor pair with the largest square number factor, we get √1104 = (√16)(√69) = 4√69 ≈ 33.2265.

1104 is the sum of the sixteen prime numbers from 37 to 103. Do you know what those prime numbers are?

1104 is also the sum of eight consecutive primes and two consecutive primes:
113 + 127 + 131 + 137 + 139 + 149 +151 + 157  = 1104
547 + 557 = 1104

1103 and Level 2

The fourteen clues you see in this puzzle are all you need to find all the factors from 1 to 10 and complete the multiplication table. Can you find all those factors?

Print the puzzles or type the solution in this excel file: 10-factors-1102-1110

Here are some facts about the number 1103:

  • 1103 is a prime number.
  • Prime factorization: 1103 is prime.
  • The exponent of prime number 1103 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1103 has exactly 2 factors.
  • Factors of 1103: 1, 1103
  • Factor pairs: 1103 = 1 × 1103
  • 1103 has no square factors that allow its square root to be simplified. √1103 ≈ 33.21144

How do we know that 1103 is a prime number? If 1103 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1103 ≈ 33.2. Since 1103 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1103 is a prime number.

1103 is the sum of the nineteen prime numbers from 19 to 101.

1103 is palindrome 191 in BASE 29 because 1(29²) + 9(29) + 1(1) = 1103

1102 and Level 1

Write each number from 1 to 10 in both the first column and the top row so that those numbers are the factors of the given clues. This one is not too difficult, so if you haven’t solved one of these puzzles before, give it a try!

Print the puzzles or type the solution in this excel file: 10-factors-1102-1110

Here is some information about the number 1102:

  • 1102 is a composite number.
  • Prime factorization: 1102 = 2 × 19 × 29
  • 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 1102 has exactly 8 factors.
  • Factors of 1102: 1, 2, 19, 29, 38, 58, 551, 1102
  • Factor pairs: 1102 = 1 × 1102, 2 × 551, 19 × 58, or 29 × 38
  • 1102 has no square factors that allow its square root to be simplified. √1102 ≈ 33.19639

1102 = 2(29)(19) so we know that 480-1102-1202 is a Pythagorean triple
calculated from 29²-19², 2(29)(19), 29²+19²

1102 is also the hypotenuse of a Pythagorean triple:
760-798-1102 which is (20-21-29) times 38.

1102 is a palindrome when it is written in a couple of other bases:
It’s 2F2 in BASE 20 (F is 15 base 10) because 2(20²) + 15(20) + 2(1) = 1102,
and it’s 262 in BASE 22 because 2(22²) + 6(22) + 2(1) = 1102.

1101 and Level 6

Which common factor of 6 and 24 will help you solve this puzzle? 2, 3, or 6?
Likewise, possible common factors of 8 and 4 are 1, 2, and 4, and for 48 and 36, you must choose between 4, 6, and 12.

In each case, only one choice will work with all the other clues in the puzzle.  You can figure out the correct choices and complete the entire puzzle by using logic. Good luck!

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

Here are a few facts about the number 1101:

Since it is made with a zero and exactly three identical numbers, it is divisible by 3.

  • 1101 is a composite number.
  • Prime factorization: 1101 = 3 × 367
  • 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 1101 has exactly 4 factors.
  • Factors of 1101: 1, 3, 367, 1101
  • Factor pairs: 1101 = 1 × 1101 or 3 × 367
  • 1101 has no square factors that allow its square root to be simplified. √1101 ≈ 33.18132

1101 is a palindrome when it is written in two other bases:
It’s 373 in BASE 18 because 3(18²) + 7(18) + 3(1) = 1101,
and 1J1 in BASE 25 (J is 19 in base 10) because 25² + 19(25) + 1 = 1101

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!