Prime factorization

1152 Will You See All the Prime Factors in This Factor Tree?

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Will you see all the prime factors in this factor tree when all of the factors are the same color? You know how to count, but would you possibly not count one of the prime factors or possibly count one of them twice?

Is it easier to count the prime factors in the following factor trees?

If you happen to have two different colors of ink and/or pencils around when you make factor trees, they might be easier to read especially if the factored number has a lot of factors like 1152 does.

Here’s what I’ve learned about the number 1152:

  • 1152 is a composite number.
  • Prime factorization: 1152 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3, which can be written 1152 = 2⁷ × 3²
  • The exponents in the prime factorization are 7 and 2. Adding one to each and multiplying we get (7 + 1)(2 + 1) = 8 × 3 = 24. Therefore 1152 has exactly 24 factors.
  • Factors of 1152: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 128, 144, 192, 288, 384, 576, 1152
  • Factor pairs: 1152 = 1 × 1152, 2 × 576, 3 × 384, 4 × 288, 6 × 192, 8 × 144, 9 × 128, 12 × 96, 16 × 72, 18 × 64, 24 × 48, or 32 × 36
  • Taking the factor pair with the largest square number factor, we get √1152 = (√576)(√2) = 24√2 ≈ 33.94112

The last two digits of 1152 are 52, a number divisible by 4, so 1152 can be evenly divided by 4.
1 + 1 + 5 + 2 = 9, so 1152 is divisible by 9.

It is so easy to tell if a number can be evenly divided by 4 or 9 AND it is so easy to divide by 4 or by 9. When I make a factor cake, I like to see if the current layer of the cake is divisible by 4 or by 9 before I check to see if it is divisible by a prime number.

From that cake, I can quickly tell that 1152 = 2⁷ × 3² by simply counting by 2’s to find the powers of 2 and 3. All the numbers being the same color doesn’t even slow me down.

I can also easily find the √1152 by taking the square root of everything on the outside of the cake:
√1152  = √(4·4) · (√4)(√9)(√2) = (4·2·3)√2 = 24√2

Since MOST square roots that can be simplified are divisible by 4, or by 9, or by both, this is a good strategy to find their square roots.

Here are some more facts about this number:

1152 is the sum of the fourteen prime numbers from 53 to 109,
and it is the sum of the twelve prime numbers from 71 to 127.

34² – 2² = 1152 so we are only 2² = 4 numbers away from the next perfect square.

1152 looks interesting when it is written in these bases:
It’s 800 in BASE 12 because 8(12²) = 1152,
242 in BASE 23 because 2(23²) + 4(23) + 2(1) = 1152
200 in BASE 24 because 2(24²) = 1152,
WW in BASE 35 (W is 32 base 10) because 32(35) + 32(1) = 32(36) = 1152, and
it’s W0 in BASE 36 because 32(36) = 1152

1150 Perfectly Symmetrical Factor Trees

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Factor trees can look lovely if they have symmetrical branches. 1150 can make that kind of a tree:

The same number can also make a more disorderly-looking tree:

All of those trees are correct factor trees. And several more can still be made for the number 1150.

What else can I tell you about that number?

  • 1150 is a composite number.
  • Prime factorization: 1150 = 2 × 5 × 5 × 23, which can be written 1150 = 2 × 5² × 23
  • 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 1150 has exactly 12 factors.
  • Factors of 1150: 1, 2, 5, 10, 23, 25, 46, 50, 115, 230, 575, 1150
  • Factor pairs: 1150 = 1 × 1150, 2 × 575, 5 × 230, 10 × 115, 23 × 50, or 25 × 46,
  • Taking the factor pair with the largest square number factor, we get √1150 = (√25)(√46) = 5√46 ≈ 33.91165

1150 is the hypotenuse of two Pythagorean triples:
690-920-1150 which is (3-4-5) times 230
322-1104-1150 which is (7-24-25) times 46

1150 looks interesting when it is written in a couple of different bases:
It’s 3232 in BASE 7 because 3(7³) + 2(7²) + 3(7) + 2(1) = 1150
and 6A6 in BASE 13 (A is 10 base 10) because 6(13²) + 10(13) + 6(1) = 1150

1149 and Level 2

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Look at a typical 1 – 10 multiplication table. There is only one column on it that has the numbers 63, 27, and 72. What column is that? Put that column number in the cell in the top row above those numbers and you will have done the first step in completing this puzzle.  You will need to write the numbers from 1 to 10 in both the first column and the top row to solve this puzzle. Can you find the correct places to put each number?

Print the puzzles or type the solution in this excel file: 10-factors-1148-1160

Here are a few facts about the number 1149:

  • 1149 is a composite number.
  • Prime factorization: 1149 = 3 × 383
  • 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 1149 has exactly 4 factors.
  • Factors of 1149: 1, 3, 383, 1149
  • Factor pairs: 1149 = 1 × 1149 or 3 × 383
  • 1149 has no square factors that allow its square root to be simplified. √1149 ≈ 33.8969

1149 is palindrome 1D1 in BASE 28 (D is 13 base 10)
because 28² + 13(28) + 1 = 1149

1148 and Level 1

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This level 1 puzzle has only one solution, and I’m sure you can find it. Just write the numbers from 1 to 10 in both the first column and the top row so that the clues and those numbers make a valid multiplication table.

Print the puzzles or type the solution in this excel file: 10-factors-1148-1160

Now here’s some information about the number 1148:

  • 1148 is a composite number.
  • Prime factorization: 1148 = 2 × 2 × 7 × 41, which can be written 1148 = 2² × 7 × 41
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1148 has exactly 12 factors.
  • Factors of 1148: 1, 2, 4, 7, 14, 28, 41, 82, 164, 287, 574, 1148
  • Factor pairs: 1148 = 1 × 1148, 2 × 574, 4 × 287, 7 × 164, 14 × 82, or 28 × 41
  • Taking the factor pair with the largest square number factor, we get √1148 = (√4)(√287) = 2√287 ≈ 33.88215

1148 is the sum of two consecutive primes:
571 + 577 = 1148

1148 is the hypotenuse of a Pythagorean triple:
252-1120-1148 which is 28 times (9-40-41)

1148 looks interesting to be in these other bases:
It’s 1120112 in BASE 3 because 3⁶ + 3⁵ + 2(3⁴) + 3² + 3¹ + 2(3⁰) = 1148,
1515 in BASE 9 because 9³ + 5(9²) + 9 + 5(1) = 1148,
and 161 in BASE 31 because 31² + 6(31) + 1 = 1148

1147 Dragonfly Challenge

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This “dragonfly” challenges you to solve the puzzle. It won’t be easy, but if you stick to using logic the entire time, you will be able to do it!

Print the puzzles or type the solution in this excel file: 12 factors 1134-1147

Here are a few facts about the number 1147:

  • 1147 is a composite number.
  • Prime factorization: 1147 = 31 × 37
  • 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 1147 has exactly 4 factors.
  • Factors of 1147: 1, 31, 37, 1147
  • Factor pairs: 1147 = 1 × 1147 or 31 × 37
  • 1147 has no square factors that allow its square root to be simplified. √1147 ≈ 33.86739

1147 is the hypotenuse of a Pythagorean triple:
372-1085-1147 which is 31 times (12-35-37)

34² – 3² = 1147, so we are only 9 numbers away from the next perfect square.

1147 is a leg in a few Pythagorean triples including this primitive:
204-1147-1165 calculated from 2(34)(3), 34² – 3², 34² + 3²

I like the way 1147 looks when it is written in some other bases:
It’s 5151 in BASE 6 because 5(6³) + 1(6²) + 5(6) + 1(1) = 1147,
7B7 in BASE 12 (B is 11 base 10) because 7(12²) + 11(12) + 7(1) = 1147,
and VV in BASE 36 (V is 31 base 10) because 31(36) + 31(1) = 31(37) = 1147

1146 Mystery Level

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This puzzle might start off easy enough, but soon afterward you will have to consider some serious logic in order to complete it. Good luck!

Print the puzzles or type the solution in this excel file: 12 factors 1134-1147

Now let me tell you about the number 1146:

  • 1146 is a composite number.
  • Prime factorization: 1146 = 2 × 3 × 191
  • 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 1146 has exactly 8 factors.
  • Factors of 1146: 1, 2, 3, 6, 191, 382, 573, 1146
  • Factor pairs: 1146 = 1 × 1146, 2 × 573, 3 × 382, or 6 × 191
  • 1146 has no square factors that allow its square root to be simplified. √1146 ≈ 33.85262

1146 is a palindrome in two other bases:
It’s 14041 in BASE 5 because 1(5⁴) + 4(5³) + 0(5²) + 4(5¹) + 1(5⁰) = 1146,
and it’s 282 in BASE 22 because 2(22²) + 8(22) + 2(1) = 1146

1145 Mysterious Butterfly

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This adventurous butterfly invites you to solve its mystery. Be warned, it might not be as easy as it seems. Don’t write any factors down unless you are SURE they belong where you’re putting them. Let logic be your guide.

Print the puzzles or type the solution in this excel file: 12 factors 1134-1147

Here are more facts about the number 1145:

  • 1145 is a composite number.
  • Prime factorization: 1145 = 5 × 229
  • 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 1145 has exactly 4 factors.
  • Factors of 1145: 1, 5, 229, 1145
  • Factor pairs: 1145 = 1 × 1145 or 5 × 229
  • 1145 has no square factors that allow its square root to be simplified. √1145 ≈ 33.83785

28² + 19² = 1145
32² + 11² = 1145

1145 in the hypotenuse of FOUR Pythagorean triples
300-1105-1145 which is 5 times 60-221-229
423-1064-1145 calculated from 28² – 19² , 2(28)(19) , 28² + 19²
687-916-1145 which is (3-4-5) times 229
704-903-1145 calculated from 2(32)(11) , 32² – 11² , 32² + 11²

1145 is a palindrome in two bases:
It’s 515 in BASE 15 because  5(15²) + 1(15) + 5(1) = 1145, and
1I1 in BASE 26 (I is 18 base 10) because 26² + 18(26) + 1 = 1145

1144 Let’s Make a Factor Cake!

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Occasionally, I’ll put a factor cake on this blog, but until today, I haven’t really shown you how to MAKE a factor cake. I made this simple gif to show you how to make a factor cake from start to finish. You can watch it over and over again if you like. Just repeatedly divide by the smallest prime number divisor.

1144 Prime Factor Cake

make science GIFs like this at MakeaGif

Here’s more about the number 1144:

  • 1144 is a composite number.
  • Prime factorization: 1144 = 2 × 2 × 2 × 11 × 13, which can be written 1144 = 2³ × 11 × 13
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1144 has exactly 16 factors.
  • Factors of 1144: 1, 2, 4, 8, 11, 13, 22, 26, 44, 52, 88, 104, 143, 286, 572, 1144
  • Factor pairs: 1144 = 1 × 1144, 2 × 572, 4 × 286, 8 × 143, 11 × 104, 13 × 88, 22 × 52, or 26 × 44
  • Taking the factor pair with the largest square number factor, we get √1144 = (√4)(√286) = 2√286 ≈ 33.82307

1144 is obviously divisible by 11, but it isn’t so obvious that 1144 is the sum of the twenty-two prime numbers from 11 to 101.

1144 is the hypotenuse of a Pythagorean triple:
440-1056-1144 which is (5-12-13) times 88

1144 is palindrome 3223 in BASE 7 because 3(7³) + 2(7²) + 2(7) +3(1) = 1144

1143 and Level 6

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I hope you get “hooked” on factoring puzzles after you solve this one! It won’t be easy, but if you stick with it, you will be able to solve it. Good luck!

Print the puzzles or type the solution in this excel file: 12 factors 1134-1147

Now I’ll share some facts about the number 1143:

  • 1143 is a composite number.
  • Prime factorization: 1143 = 3 × 3 × 127, which can be written 1143 = 3² × 127
  • 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 1143 has exactly 6 factors.
  • Factors of 1143: 1, 3, 9, 127, 381, 1143
  • Factor pairs: 1143 = 1 × 1143, 3 × 381, or 9 × 127
  • Taking the factor pair with the largest square number factor, we get √1143 = (√9)(√127) = 3√127 ≈ 33.80828

1143 is repdigit 333 in BASE 19 because 3(19² + 19 + 1) = 3(381) = 1143

1142 and Level 5

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If you start in the right place, you will be able to solve this puzzle using logic and basic multiplication facts. Don’t get discouraged, if you make a mistake. I know you can figure this one out!

Print the puzzles or type the solution in this excel file: 12 factors 1134-1147

Here are some facts about the number 1142:

  • 1142 is a composite number.
  • Prime factorization: 1142 = 2 × 571
  • 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 1142 has exactly 4 factors.
  • Factors of 1142: 1, 2, 571, 1142
  • Factor pairs: 1142 = 1 × 1142 or 2 × 571
  • 1142 has no square factors that allow its square root to be simplified. √1142 ≈ 33.79349

1142 is a palindrome in three different bases:
It’s 949 in BASE 11 because 9(11²) + 4(11) + 9(1) = 1142,
3G3 in BASE 17 (G is 16 base 10) because 3(17²) + 16(17) + 3(1) = 1142,
and 2H2 in BASE 20 (H is 17 base 10) because 2(20²) + 17(20) + 2(1) = 1142