factor pairs

1125 May I Interest You In a Little Cake?

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1 + 1 + 2 + 5 = 9, so 1125 can be evenly divided by both 3 and 9.
1125 can also be evenly divided by 5 because the last digit can be divided by 5.
1125 can be evenly divided by 25 because the last two digits can be divided by 25.
1125 can be evenly divided by 125 because the last three digits can be divided by 125.

May I interest you in a little cake? Although I’ve put many factor trees on this website, I actually prefer making factor cakes. I like them because all the prime factors can easily be found IN ORDER on the outside of the cake. The more layers the factor cake has, the more I like it. Here is THE factor cake for 1125:

As you can see, the prime factorization of 1125 is 3 × 3 × 5 × 5 × 5. All of its prime factors are written nicely in order. None of them are hiding anyplace like they might in a factor tree.

Here are some facts about the number 1125:

  • 1125 is a composite number.
  • Prime factorization: 1125 = 3 × 3 × 5 × 5 × 5, which can be written 1125 = 3² × 5³
  • The exponents in the prime factorization are 2 and 3. Adding one to each and multiplying we get (2 + 1)(3 + 1) = 3 × 4 = 12. Therefore 1125 has exactly 12 factors.
  • Factors of 1125: 1, 3, 5, 9, 15, 25, 45, 75, 125, 225, 375, 1125
  • Factor pairs: 1125 = 1 × 1125, 3 × 375, 5 × 225, 9 × 125, 15 × 75, or 25 × 45
  • Taking the factor pair with the largest square number factor, we get √1125 = (√225)(√5) = 15√5 ≈ 33.54102

1125 is the hypotenuse of three Pythagorean triples:
675-900-1125 which is (3-4-5) times 225,
315-1080-1125 which is (7-24-25) times 45, and
396-1053-1125 which is 9 times (44-117-125)

1125 looks interesting to me when it is written in these bases:
It’s 5A5 in BASE 14 (A is 10 base 10) because 5(14²) + 10(14) + 5(1) = 1125
500 in BASE 15 because 5(15²) = 1125, and
it’s 3F3 in BASE 17 (F is 15 base 10) because 3(17²) + 15(17) + 3(1) = 1125

1124 and Level 3

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The common factors of 54 and 60 are 1, 2, 3, and 6. Just one of those common factors will put only numbers from 1 to 10 in the top row. That’s the factor you need to choose. To complete the puzzle, all the numbers from 1 to 10 must go in both the first column and the top row. Can you solve this puzzle?

Print the puzzles or type the solution in this excel file: 10-factors-1121-1133

Here are a few facts about the number 1124:

  • 1124 is a composite number.
  • Prime factorization: 1124 = 2 × 2 × 281, which can be written 1124 = 2² × 281
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 1124 has exactly 6 factors.
  • Factors of 1124: 1, 2, 4, 281, 562, 1124
  • Factor pairs: 1124 = 1 × 1124, 2 × 562, or 4 × 281
  • Taking the factor pair with the largest square number factor, we get √1124 = (√4)(√281) = 2√281 ≈ 33.52611

1124 is the hypotenuse of a Pythagorean triple:
640-924-1124 which is 4 times 160-231-281

If I asked you to tell me what is significant about this set of numbers {13, 16, 19, 22}, what would you say?

Perhaps you would tell me they make an arithmetic sequence in which the common difference is 3.

What you probably wouldn’t tell me is that 1124 is a palindrome in those four bases!
It’s 686 in BASE 13 because 6(13²) + 8(13) + 6(1) = 1124,
464 in BASE 16 because 4(16²) + 6(16) + 4(1) = 1124
323 in BASE 19 because 3(19²) + 2(19) + 3(1) = 1124, and
272 in BASE 22 because 2(22²) + 7(22) + 2(1) = 1124

1122 Factor Trees

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We can easily tell that 1122 is divisible by 2 because it is even.
It is divisible by 3 because 1 + 2 = 3, no matter how many times you do it.
11 and 22 are both divisible by 11, so obviously, 1122 is as well.
1122 is also divisible by 17, but there isn’t an easy trick to use to find that out without just dividing it.

There are 24 ways to arrange the four factors of 1122. Each of those 24 ways can be the bottom four numbers of a factor tree (See the first three trees below.) or the numbers running down the side of a factor tree (See the last four trees below.).

1122 makes some very well proportioned factor trees!

  • 1122 is a composite number.
  • Prime factorization: 1122 = 2 × 3 × 11 × 17
  • 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 1122 has exactly 16 factors.
  • Factors of 1122: 1, 2, 3, 6, 11, 17, 22, 33, 34, 51, 66, 102, 187, 374, 561, 1122
  • Factor pairs: 1122 = 1 × 1122, 2 × 561, 3 × 374, 6 × 187, 11 × 102, 17 × 66, 22 × 51, or 33 × 34
  • 1122 has no square factors that allow its square root to be simplified. √1122 ≈ 33.49627

1122 is the product of two consecutive number: 33 × 34 = 1122
That makes 1122 the sum of the first 33 even numbers:
2 + 4 + 6 + 8 + . . . + 62 + 64 + 66 = 1122

1122 is the hypotenuse of a Pythagorean triple:
528-990-1122 which is (8-15-17) times 66

1122 is palindrome 2G2 in BASE 20 (G is 16 base 10)
because 2(20²) + 16(20) + 2(1) = 1122

1121 and Level 1

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If you’ve learned how to multiply and divide, then you can solve this puzzle. Just write the numbers from 1 to 10 in both the first column and the top row so that the clues and those factors make a multiplication table. You can definitely do this one!

Print the puzzles or type the solution in this excel file: 10-factors-1121-1133

Here are a few facts about the number 1121:

  • 1121 is a composite number.
  • Prime factorization: 1121 = 19 × 59
  • 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 1121 has exactly 4 factors.
  • Factors of 1121: 1, 19, 59, 1121
  • Factor pairs: 1121 = 1 × 1121 or 19 × 59
  • 1121 has no square factors that allow its square root to be simplified. √1121 ≈ 33.48134

1121 is a palindrome in two other bases:
It’s 1C1 in BASE 28 (C is 12 base 10) because 28² + 12(28) + 1 = 1121,
and it’s 131 in BASE 32 because 32² + 3(32) + 1 = 1121

1120 There’s Lots of Shade under These Factor Trees!

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It’s hot this summer, but you can rest under the shade of some of 1120’s factor trees. Whether you want a tall tree or a wide one, these are only a few of the MANY possible ones you can choose. Every one of them gives you the same prime factors which are shown in pink here.

Here are some things I’ve learned about the number 1120:

  • 1120 is a composite number.
  • Prime factorization: 1120 = 2 × 2 × 2 × 2 × 2 × 5 × 7, which can be written 1120 = 2⁵ × 5 × 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) = 7 × 2 × 2 = 28. Therefore 1120 has exactly 28 factors.
  • Factors of 1120: 1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 28, 32, 35, 40, 56, 70, 80, 112, 140, 160, 224, 280, 560, 1120
  • Factor pairs: 1120 = 1 × 1120, 2 × 560, 4 × 280, 5 × 224, 7 × 160, 8 × 140, 10 × 112, 14 × 80, 16 × 70, 20 × 56, 28 × 40, or 32 × 35
  • Taking the factor pair with the largest square number factor, we get √1120 = (√16)(√70) = 4√70 ≈ 33.4664

1120 is the sum of the twenty prime numbers from 17 to 101.
It is also the sum of these two consecutive primes:
557 + 563 = 1120

1120 is the hypotenuse of exactly one Pythagorean triple:
672-896-1120 which is (3-4-5) times 224

I like the way 1120 looks in these other bases:
It’s 1112111 in BASE 3 because 3⁶ + 3⁵ + 3⁴ + 2(3³) + 3² + 3  + 1 = 1120,
929 in BASE 11 because 9(11²) + 2(11) + 9(1) = 1120,
WW in BASE 34 (W is 32 base 10) because 32(34) + 32(1) = 32(35) = 1120,
and W0 in BASE 35 because 32(35) = 1120

1119 Mystery

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Will this Mystery Level Puzzle be easy or difficult? That is part of the mystery! You might find the logic needed to solve it a mystery, too, but I assure you it can be solved without guessing and checking. Can you do it?

Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

Here are some facts about the number 1119:

  • 1119 is a composite number.
  • Prime factorization: 1119 = 3 × 373
  • 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 1119 has exactly 4 factors.
  • Factors of 1119: 1, 3, 373, 1119
  • Factor pairs: 1119 = 1 × 1119 or 3 × 373
  • 1119 has no square factors that allow its square root to be simplified. √1119 ≈ 33.45146

1119 is the sum of three consecutive prime numbers, one of which is its prime factor:
367 + 373 + 379 = 1119

1119 is the hypotenuse of a Pythagorean triple:
756-825-1119 which is 3 times (252-275-373)

1119 is a palindrome in two other bases:
It’s 383 in BASE 18 because 3(18²) + 8(18) + 3(1) = 1119, and
1H1 in BASE 26 (H is 17 base 10) because 1(26²) + 17(26) + 1(1) = 1119

1118 and Level 6

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Each Level 6 puzzle has different logic needed to solve it. For this one, you will ask yourself over and over again, “Where is the only place ___ can go in the first column (or the top row)?” Just stick with it until all the numbers from 1 to 12 are in both of those places.

Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

Now here’s a little about the number 1118:

  • 1118 is a composite number.
  • Prime factorization: 1118 = 2 × 13 × 43
  • 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 1118 has exactly 8 factors.
  • Factors of 1118: 1, 2, 13, 26, 43, 86, 559, 1118
  • Factor pairs: 1118 = 1 × 1118, 2 × 559, 13 × 86, or 26 × 43
  • 1118 has no square factors that allow its square root to be simplified. √1118 ≈ 33.43651

1118 is the hypotenuse of a Pythagorean triple:
430-1032-1118 which is (5-12-13) times 86

1118 is also a leg in one of this year’s Pythagorean triples!
1118-1680-2018 calculated from 2(43)(13), 43² – 13², 43² + 13²

1117 and Level 5

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Can you solve this Level 5 puzzle or will it make you feel like toast? Seriously, I think it’s a little easier than the previous puzzle, so give it a try. You might just become the toast of the town!

Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

Now I’ll share some information about the number 1117:

  • 1117 is a prime number.
  • Prime factorization: 1117 is prime.
  • The exponent of prime number 1117 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1117 has exactly 2 factors.
  • Factors of 1117: 1, 1117
  • Factor pairs: 1117 = 1 × 1117
  • 1117 has no square factors that allow its square root to be simplified. √1117 ≈ 33.42155

How do we know that 1117 is a prime number? If 1117 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1117 ≈ 33.4. Since 1117 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1117 is a prime number.

26² + 21² = 1117

1117 is the hypotenuse of a Pythagorean triple:
235-1092-1117 calculated from 26² – 21², 2(26)(21), 26² + 21²

Here’s another way we know that 1117 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 26² + 21² = 1117 with 26 and 21 having no common prime factors, 1117 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1117 ≈ 33.4. Since 1117 is not divisible by 5, 13, 17, or 29, we know that 1117 is a prime number.

1117 is palindrome 151 in BASE 31 because 1(31²) + 5(31) + 1(1) = 1117

1116 Can I Trick You With This Puzzle?

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This is only a Level 4 puzzle, but can I trick you into making some assumptions about the puzzle’s factors? I bet I will trick a least a few people. Will you be one of them? If you don’t write a factor unless you are 100% sure it goes where you are putting it, I shouldn’t be able to trick you. Good luck with this one!

Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

Now I’ll share some facts about the number 1116:

  • 1116 is a composite number.
  • Prime factorization: 1116 = 2 × 2 × 3 × 3 × 31, which can be written 1116 = 2² × 3² × 31
  • 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 1116 has exactly 18 factors.
  • Factors of 1116: 1, 2, 3, 4, 6, 9, 12, 18, 31, 36, 62, 93, 124, 186, 279, 372, 558, 1116
  • Factor pairs: 1116 = 1 × 1116, 2 × 558, 3 × 372, 4 × 279, 6 × 186, 9 × 124, 12 × 93, 18 × 62 or 31 × 36
  • Taking the factor pair with the largest square number factor, we get √1116 = (√36)(√31) = 6√31 ≈ 33.40659

1116 looks interesting when it is represented in these other bases:
It’s 13431 in BASE 5 because 1(5⁴) + 3(5³) + 4(5²) + 3(5) + 1(1) = 1116,
VV in BASE 35 (V is 31 base 10) because 31(35) + 31(1) = 31(36) = 1116, and
V0 in BASE 36 because 31(36) = 1116

 

1115 and Level 3

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You will have to know the 11 and 12 times tables to solve this Level 3 Find the Factors 1-12 puzzle, but I’m sure you can do it! Stick with it, and don’t give up! Start with the clues at the top of the puzzle and work down row by row until it’s completed.

Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

1115 is not a clue just the puzzle number. Here are some facts about the number 1115:

  • 1115 is a composite number.
  • Prime factorization: 1115 = 5 × 223
  • 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 1115 has exactly 4 factors.
  • Factors of 1115: 1, 5, 223, 1115
  • Factor pairs: 1115 = 1 × 1115 or 5 × 223
  • 1115 has no square factors that allow its square root to be simplified. √1115 ≈ 33.39162

1115 is the sum of nine consecutive prime numbers:
103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 = 1115

1115 is the hypotenuse of a Pythagorean triple:
669-892-1115 which is (3-4-5) times 223

1115 is palindrome 2B2 in BASE 21 (B is 11 base 10)
because 2(21²) + 11(21) + 2(1) = 1115