Factors

What Kind of Shape is 1156 In?

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One of the shapes that 1156 is in is a pentagon. Below you see 1156 tiny squares arranged into a pentagon.

If you study the graphic above you will also see that 1156 is 1 more than five times the 21st triangular number.

The first few centered pentagonal numbers are 1, 6, 16, 31, 51, 76, 106. Double any centered pentagonal number 106 or greater, and you will get a number that can be represented as 552 in some base even if it’s a base nobody uses. Let’s see it at work here:
1156 × 2 = 2312, and 2312 is 552 in BASE 21

Why is that true? Because of the formula that calculates centered pentagonal numbers. For example, we know 1156 is the 22nd centered pentagonal number from
(5(21²) + 5(21) + 2(1))/2 = 1156

Written another way it looks like 1 more than 5 times the 21st triangular number:
5(21)(22)/2 + 1 = 1156

1156 is also a perfect square!

Not only is it a perfect square but it is the square of a composite number. The following two squares show how it can be made with multiple squares of its prime factors.

  • 1156 is a composite number.
  • Prime factorization: 1156 = 2 × 2 × 17 × 17, which can be written 1156 = 2²× 17²
  • 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 1156 has exactly 9 factors.
  • Factors of 1156: 1, 2, 4, 17, 34, 68, 289, 578, 1156
  • Factor pairs: 1156 = 1 × 1156, 2 × 578, 4 × 289, 17 × 68, or 34 × 34
  • 1156 is a perfect square. √1156 = 34

Another shape where you can find 1156 are these two Pythagorean triple triangles:
644-960-1156 and 544-1020-1156

1156 is also the sum of the twenty-four prime numbers from 5 to 101.

1156 is palindrome 1120211 in BASE 3 because 3⁶+3⁵+2(3⁴)+2(3²)+3+1=1156

Square number 1156 also looks like some other square numbers when it is written in these other bases:
It’s 961 in BASE 11 because 9(11²) + 6(11) + 1(1) = 1156,
484 in BASE 16 because 4(16²) + 8(16) + 4(1) = 1156,
400 in BASE 17 because 4(17²) = 1156,
169 in BASE 31 because 31² + 6(31) + 9(1) = 1156,
144 in BASE 32 because 32² + 4(32) + 4(1) = 1156,
121 in BASE 33 because 33² + 2(33) + 1(1) = 1156,
and 100 in BASE 34 because 34² = 1156

 

Reasons to Celebrate 1155

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1155 is the product of the four smallest odd prime numbers. That alone is a reason to celebrate. Here, I’ve made a factor cake with 1155 for you to enjoy!

1155 Factor Cake

 

make science GIFs like this at MakeaGif

Besides 1155 being the smallest number divisible by the first 4 odd prime numbers, what else can I say about it?

  • 1155 is a composite number.
  • Prime factorization: 1155 = 3 × 5 × 7 × 11
  • 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 1155 has exactly 16 factors.
  • Factors of 1155: 1, 3, 5, 7, 11, 15, 21, 33, 35, 55, 77, 105, 165, 231, 385, 1155
  • Factor pairs: 1155 = 1 × 1155, 3 × 385, 5 × 231, 7 × 165, 11 × 105, 15 × 77, 21 × 55, or 33 × 35
  • 1155 has no square factors that allow its square root to be simplified. √1155 ≈ 33.98529

Since it’s number of prime factors is a power of 2, it can make some very well-balanced and attractive factor trees:

1155 is the hypotenuse of a Pythagorean triple:
693-924-1155 which is (3-4-5) times 231

1155 is one number away from the next perfect square, AND it is the difference of two squares EIGHT different ways.
34² – 1² = 1155
38² – 17² = 1155
46² – 31² = 1155
58² – 47² = 1155
86² – 79² = 1155
118² – 113² = 1155
194² – 191² = 1155
578² – 577² = 1155

Those eight ways correspond to its eight factor pairs. Notice that both of the numbers in all of its factor pairs are odd numbers.

I like the way 1155 looks in some other bases:
It’s 3A3 in BASE 18 (A is 10 base 10) because 3(18²) + 10(18) + 3(1) = 1155
XX in BASE 34 (X is 33 base 10) because 33(34) + 33(1) = 33(35) = 1155
X0 in BASE 35 because 33(35) = 1155

XX X0 . . . Hugs and Kisses. You gotta love that!

1154 and Level 5

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I’m sure you can have a lot of fun solving this puzzle. Remember to use logic before you write down any of the factors, and it will be fun instead of frustrating.

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

Now I’ll share some facts about the number 1154:

  • 1154 is a composite number.
  • Prime factorization: 1154 = 2 × 577
  • 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 1154 has exactly 4 factors.
  • Factors of 1154: 1, 2, 577, 1154
  • Factor pairs: 1154 = 1 × 1154 or 2 × 577
  • 1154 has no square factors that allow its square root to be simplified. √1154 ≈ 33.97058

1154 is the sum of eight consecutive prime numbers:
127 + 131 + 137 + 139 + 149 + 151 + 157 + 163 = 1154

25² + 23² = 1154

1154 is the hypotenuse of a Pythagorean triple:
96-1150-1154 calculated from 25² – 23², 2(25)(23), 25² + 23²

1154 is palindrome 202 in BASE 24
because 2(24²) + 2(1) = 2(24² + 1) = 2(577) = 1154

Did you notice that 1154 has a relationship with 23², 24², and 25²?

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

1151 and Level 3

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If you know the common prime factor for 27 and 30, then you can at least start this puzzle. If you work down the first column cell by cell using logic, you should be able to solve the puzzle, too. Good luck!

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

Here are some facts about the number 1151:

  • 1151 is a prime number.
  • Prime factorization: 1151 is prime.
  • The exponent of prime number 1151 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1151 has exactly 2 factors.
  • Factors of 1151: 1, 1151
  • Factor pairs: 1151 = 1 × 1151
  • 1151 has no square factors that allow its square root to be simplified. √1151 ≈ 33.92639

How do we know that 1151 is a prime number? If 1151 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1151 ≈ 33.9. Since 1151 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1151 is a prime number.

1151 is the sum of consecutive primes three different ways:
It is the sum of the twenty-three prime numbers from 7 to 101.
223 + 227 + 229 + 233 + 239 = 1151 and
379 + 383 + 389 = 1151

OEIS.org states that 1151 is the smallest number that is the sum of consecutive prime numbers four different ways, I think they must be considering 1151 = 1151 to be one of those ways.

1151 is palindrome 1L1 in BASE 25 (L is 21 base 10)
because 25² + 21(25) + 1 = 1151

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