Prime factorization

STOP! Look How Cool a Number 1080 Is!

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STOP Sign Geometry Fact:

What can I tell you about the number 1080?  Lots of things!

The sum of the interior angles of an eight-sided polygon such as a stop sign is 1080°.

The sum of the interior angles of an octagon is 6(180°) = 1080°. Convex or Concave, it doesn’t matter, the sum of those interior angles of an eight-sided polygon will still be 1080°, as illustrated below:

Factor Trees for 1080:

Here are a couple of the MANY possible factor trees for 1080:

Factors of 1080:

There is only one number less than 1080 that has as many factors as 1080 does. What was that number? 840. How many factors does1080 have? 32. Wow!

  • 1080 is a composite number.
  • Prime factorization: 1080 = 2 × 2 × 2 × 3 × 3 × 3 × 5, which can be written 1080 = 2³ × 3³ × 5.
  • The exponents in the prime factorization are 3, 3, and 1. Adding one to each and multiplying we get (3 + 1)(3 + 1)(1 + 1) ) = 4 × 4 × 2 = 32. Therefore 1080 has exactly 32 factors.
  • Factors of 1080: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 27, 30, 36, 40, 45, 54, 60, 72, 90, 108, 120, 135, 180, 216, 270, 360, 540, 1080
  • Factor pairs: 1080 = 1 × 1080, 2 × 540, 3 × 360, 4 × 270, 5 × 216, 6 × 180, 8 × 135, 9 × 120, 10 × 108, 12 × 90, 15 × 72, 18 × 60, 20 × 54, 24 × 45, 27 × 40 or 30 × 36
  • Taking the factor pair with the largest square number factor, we get √1080 = (√36)(√30) = 6√30 ≈ 32.86335

1080 has 28 composite factors and is the smallest number that can make that claim. (Of its 32 factors, all are composite numbers except 1, 2, 3, and 5). That’s more than 840’s 27 composite factors. (Its 32 factors minus 1, 2, 3, 5, and 7)

Sum Difference Puzzles:

30 has four factor pairs. One of those pairs adds up to 13, and  another one subtracts to 13. Put the factors in the appropriate boxes in the first puzzle.

1080 has sixteen factor pairs. One of the factor pairs adds up to ­78, and a different one subtracts to 78. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

Other Facts about the Number 1080:

1080 is the sum of these four consecutive prime numbers:
263 + 269 + 271 + 277 = 1080

1080 is the sum of four consecutive powers of three:
3⁶ + 3⁵ + 3⁴ + 3³ = 1080

1080 is the hypotenuse of a Pythagorean triple:
648-864-1080 which is (3-4-5) times 216

Note that 5(6³) = 5(216) = 1080 so 1080 is 500 in BASE 6.
It’s palindrome 252 in BASE 22 because 2(22²) + 5(22) + 2(1) = 1080,
UU in BASE 35 (U is 30 base 10) because 30(35) + 30(1) = 30(36) = 1080,
and it’s U0 in BASE 36 because 30(36) = 1080

And now I’ll STOP writing about how cool 1080 is.

1079 An Easier Find The Factors Challenge?

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This find the factors challenge puzzle might be easier than most of the challenge puzzles are, but it will still give you plenty of reasons to think about what factors you should put where. You need to put all the numbers from 1 to 10 in each of the four bold areas so that those numbers are the factors of the given clues. There is only one solution. Can you find it?

Print the puzzles or type the solution in this excel file: 10-factors-1073-1079

Here’s a little about the number 1079:

  • 1079 is a composite number.
  • Prime factorization: 1079 = 13 × 83
  • 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 1079 has exactly 4 factors.
  • Factors of 1079: 1, 13, 83, 1079
  • Factor pairs: 1079 = 1 × 1079 or 13 × 83
  • 1079 has no square factors that allow its square root to be simplified. √1079 ≈ 32.84814

1079 is the sum of three consecutive prime numbers:
353 + 359 + 367 = 1079

1079 is also the hypotenuse of a Pythagorean triple:
415-996-1079 which is (5-12-13) times 83

1078 and Level 6

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This Level 6 puzzle might be just a little bit easier than usual, so if you’ve never done this level before, be sure to try this one! You can succeed if you stick with it!

Print the puzzles or type the solution in this excel file: 10-factors-1073-1079

Here are a few facts about the number 1078:

  • 1078 is a composite number.
  • Prime factorization: 1078 = 2 × 7 × 7 × 11, which can be written 1078 = 2 × 7² × 11
  • 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 1078 has exactly 12 factors.
  • Factors of 1078: 1, 2, 7, 11, 14, 22, 49, 77, 98, 154, 539, 1078
  • Factor pairs: 1078 = 1 × 1078, 2 × 539, 7 × 154, 11 × 98, 14 × 77, or 22 × 49,
  • Taking the factor pair with the largest square number factor, we get √1078 = (√49)(√22) = 7√22 ≈ 32.83291

1 – 0 + 7 – 8 = 0 so 1078 is divisible by 11.

Since 11 is its largest prime factor we can make a lovely factor cake for 1078:

 

1078 is palindrome 4554 in BASE 6 because 4(6³) + 5(6²) + 5(6) + 4(1) = 1078

1077 and Level 5

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If you aren’t careful I might trick you into writing the numbers from 1 to 10 in the wrong places on this level 5 puzzle. Don’t let me trick you! Only write a factor if you know for sure where it goes. Study all the clues until logic directs you where to start.

Print the puzzles or type the solution in this excel file: 10-factors-1073-1079

Here is a little information about the number 1077:

It was difficult finding something unique about 1077, so I’m writing about a few things that I don’t usually mention:

1077 can be written as the difference of two squares two different ways:
539² – 538² = 1077
181² – 178² = 1077

1077 is the sum of two consecutive numbers 538 + 539 = 1077
1077 is the sum of three consecutive numbers 358 + 359 + 360
1077 is also the sum of three consecutive odd numbers 357-359-361

1077 is a leg in four Pythagorean triples:
1077-1436-1795 which is (3-4-5) times 359,
1077-193320-193323 which is 3 times (359-64440-64441),
1077-64436-64445 calculated from 181² – 178², 2(181)(178), 181² + 178², and
1077-579964-579965 calculated from 539² – 538², 2(539)(538), 539² + 538²

1076 and Level 4

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If you roll a pair of dice, you are taking a chance that the roll might not be favorable for you. Don’t take your chances when solving this puzzle. It can be solved completely by relying on logic. It won’t require any luck, but “Good luck!” anyway.

Print the puzzles or type the solution in this excel file: 10-factors-1073-1079

Here is some information about the number 1076:

  • 1076 is a composite number.
  • Prime factorization: 1076 = 2 × 2 × 269, which can be written 1076 = 2² × 269
  • 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 1076 has exactly 6 factors.
  • Factors of 1076: 1, 2, 4, 269, 538, 1076
  • Factor pairs: 1076 = 1 × 1076, 2 × 538, or 4 × 269
  • Taking the factor pair with the largest square number factor, we get √1076 = (√4)(√269) = 2√269 ≈ 32.80244

1076 is the hypotenuse of a Pythagorean triple:
276-1040-1076 which is 4 times (69-260-269)

1076 is a palindrome in a couple of bases:
It’s 434 in BASE 16 because 4(16²) + 3(16) +4(1) = 1076, and
it’s 1I1 in BASE 25 (I is 18 base 10) because 25² + 18(25) + 1 = 1076

1075 and Level 3

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Begin at the top of this level 3 puzzle and find the factors clue by clue until you reach the bottom. You can solve this puzzle and have fun doing it!

Print the puzzles or type the solution in this excel file: 10-factors-1073-1079

Here are some facts about the number 1075:

  • 1075 is a composite number.
  • Prime factorization: 1075 = 5 × 5 × 43, which can be written 1075 = 5² × 43
  • 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 1075 has exactly 6 factors.
  • Factors of 1075: 1, 5, 25, 43, 215, 1075
  • Factor pairs: 1075 = 1 × 1075, 5 × 215, or 25 × 43
  • Taking the factor pair with the largest square number factor, we get √1075 = (√25)(√43) = 5√43 ≈ 32.78719

If you had 43 quarters, you would have $10.75.
If you had 215 nickles, you would also have $10.75.

1075 is the hypotenuse of two Pythagorean triples:
645-860-1075 which is (3-4-5) times 215
301-1032-1075 which is (7-24-25) times 43

1075 is a palindrome in two consecutive bases:
It’s 898 in BASE 11 because 8(121) + 9(11) + 8(1) = 1075, and
it’s 757 in BASE 12 because 7(144) + 5(12) + 7(1) = 1075

1074 and Level 2

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Jump up and cheer! You can solve this puzzle! Simply write all the numbers from 1 to 10 in both the first column and the top row so that those numbers are the factors of the given clues.

Print the puzzles or type the solution in this excel file: 10-factors-1073-1079

Now I’ll tell you something about the number 1074:

  • 1074 is a composite number.
  • Prime factorization: 1074 = 2 × 3 × 179
  • 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 1074 has exactly 8 factors.
  • Factors of 1074: 1, 2, 3, 6, 179, 358, 537, 1074
  • Factor pairs: 1074 = 1 × 1074, 2 × 537, 3 × 358, or 6 × 179
  • 1074 has no square factors that allow its square root to be simplified. √1074 ≈ 32.77194

1074 is the short leg in these Pythagorean triples:
1074-1432-1790 which is (3-4-5) times 358
1074-32032-32050, a primitive calculated from 2(179)(3), 179² – 3², 179² + 3²
1074-96120-96126 which is 6 times (179-16020-16021)
1074-288368-288370 calculated from 2(537)(1), 537² – 1², 537² + 1²

1074 is a palindrome when it is written in two different bases:
It”s 3C3 in BASE 17 (C is 12 base 10) because 3(17²) + 12(17) + 3(1) = 1074,
and 181 in BASE 29 because 29² + 8(29) + 1 = 1074

1073 and Level 1

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You might not recognize it, but this puzzle is just a multiplication table. The factors in the table are all missing, and they aren’t in their usual places, but you have everything you need here to find the factors from 1 to 10 and then complete the entire multiplication table. It’s a level 1 puzzle so I am absolutely sure you can solve it!

Print the puzzles or type the solution in this excel file: 10-factors-1073-1079

Here is a little information about the number 1073:

  • 1073 is a composite number.
  • Prime factorization: 1073 = 29 × 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 1073 has exactly 4 factors.
  • Factors of 1073: 1, 29, 37, 1073
  • Factor pairs: 1073 = 1 × 1073 or 29 × 37
  • 1073 has no square factors that allow its square root to be simplified. √1073 ≈ 32.75668

32² + 7² = 1073
28² + 17² = 1073

Because 29 × 37 = 1073, it is the hypotenuse of FOUR Pythagorean triples, two of which are primitives:
348-1015-1073 which is 29 times (12-35-37)
740-777-1073 which is (20-21-29) times 37
448-975-1073 calculated from 2(32)(7), 32² – 7², 32² + 7²
495-952-1073 calculated from 28² – 17², 2(28)(17), 28² + 17²

1073 looks interesting when it is written in some other bases:
It’s 4545 in BASE 6 because 4(6³) + 5(6²) + 4(6) + 5(1) = 1073,
292 in BASE 21 because 2(21²) + 9(21) + 2(1) = 1073, and
TT in BASE 36 (T is 29 base 10) because 29(36) + 29(1) = 29(37) = 1073

1072 Find the Factors Challenge

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These Find the Factors Challenge puzzles are tougher than my other puzzles, but they can still be solved using logic and basic multiplication and division facts. Go ahead, give it a try! I sincerely hope you will succeed!

Print the puzzles or type the solution in this excel file: 12 factors 1063-1072

Now I’ll share some facts about the number 1072:

  • 1072 is a composite number.
  • Prime factorization: 1072 = 2 × 2 × 2 × 2 × 67, which can be written 1072 = 2⁴ × 67
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1072 has exactly 10 factors.
  • Factors of 1072: 1, 2, 4, 8, 16, 67, 134, 268, 536, 1072
  • Factor pairs: 1072 = 1 × 1072, 2 × 536, 4 × 268, 8 × 134, or 16 × 67
  • Taking the factor pair with the largest square number factor, we get √1072 = (√16)(√67) = 4√67 ≈ 32.74141

1072 is also palindrome 646 in BASE 13 because 6(13²) + 4(13) + 6(1) = 1072

1071 What I Bought at the School Book Fair

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Our elementary school recently had a book fair. I purchased three books, and I’d like to tell you a little bit about each of them.

Optical Illusions by Gianni A. Sarcone and Marie-Jo Waeber will make a nice addition to my other books on optical illusions. This one is special not only because it has a moveable design on the front cover, but also because it includes instructions on how kids or adults can make their own optical illusions. Mathematics and art often both play a part when an optical illusion is created. There is so much information in this book that I haven’t read it completely yet, but I like what I’ve read so far.

When Sophie Thinks She Can’t by Molly Bang brings up several subjects including math anxiety, bullying, and growth mindset. It also introduces tangram puzzles and making rectangles from 12 squares. Anytime you feel inclined to say, “I can’t!” add the very important word, “yet.” I like this book a lot and read it to a class of 5th graders right before I introduced them to the Find the Factors puzzles. (As I handed out the puzzles, one student with a sense of humor called out, “I can’t do these puzzles.” I smiled and said, “That was a good one,” and then proceeded to make sure every kid in the class could at least do the first few levels of the puzzle.)

All Year Round by Susan B. Katz with cute pictures by Eiko Ojala. This book uses appealing rhymes to intertwine two important early mathematical concepts: the calendar and simple geometric shapes. Some of the shapes even introduce concepts in solid geometry.

At first, I didn’t buy this book because a few things disappointed me:

  1.  It seems to imply that flowers with four round petals are shaped like a square. (The flowers are in a square box, but the focus in on the flowers, not the box.)
  2. Baseball diamonds are actually squares. Turning it on its corner does not change it from a square to a diamond, or rhombus, as it is called in mathematics. (I decided to forgive this because every square is technically a rhombus even though not every rhombus is a square. And besides, a baseball playing field really is called a baseball diamond.)
  3. Unless you cut off the rounded top of a slice of pumpkin pie, you won’t really have a triangle shape; you will have a sector. Yes, sectors look a lot like triangles, but they are not triangles.

However, after going home and thinking about it, I decided that these complaints can be good conversation starters so I went back to the book fair and bought the book. Besides, commonly-used shape names do not necessarily match correct geometric terms. I have read this book to one of my granddaughters, and we enjoyed reading and seeing how the world changes over a calendar year and identifying the simple shapes in the illustrations.

Since this is my 1071st post, I’ll share some facts about the number 1071:

  • 1071 is a composite number.
  • Prime factorization: 1071 = 3 × 3 × 7 × 17, which can be written 1071 = 3² × 7 × 17
  • 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 1071 has exactly 12 factors.
  • Factors of 1071: 1, 3, 7, 9, 17, 21, 51, 63, 119, 153, 357, 1071
  • Factor pairs: 1071 = 1 × 1071, 3 × 357, 7 × 153, 9 × 119, 17 × 63, or 21 × 51,
  • Taking the factor pair with the largest square number factor, we get √1071 = (√9)(√119) = 3√119 ≈ 32.72614

1071 is the hypotenuse of a Pythagorean triple:
504-945-1071 which is (8-15-17) times 63

OEIS.org informs us that 6³ + 7³ + 8³ = 1071, making 1071 the sum of three consecutive cubes.

1071 is consecutive digits 567 in BASE 14 because 5(14²) + 6(14) + 7(1) = 1071