Mathematics

1084 and Level 4

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Where should you put all the numbers 1 to 12 in both the top row and the first column?  You will have to think about it and use logic. Some of the clues might be tricky, but you’ll figure it all out.

Print the puzzles or type the solution in this excel file: 12 factors 1080-1086

Here are some facts about the number 1084:

  • 1084 is a composite number.
  • Prime factorization: 1084 = 2 × 2 × 271, which can be written 1084 = 2² × 271
  • 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 1084 has exactly 6 factors.
  • Factors of 1084: 1, 2, 4, 271, 542, 1084
  • Factor pairs: 1084 = 1 × 1084, 2 × 542, or 4 × 271
  • Taking the factor pair with the largest square number factor, we get √1084 = (√4)(√271) = 2√271 ≈ 32.92416

1084 is the sum of the 18 prime numbers from 23 to 101.

It is also the sum of six consecutive prime numbers:
167 + 173 + 179 + 181 + 191 + 193  = 1084

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

1056 How to Tile a 32 × 33 Floor

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32 × 33 = 1056, and OEIS.org informs us that those are the smallest rectangular dimensions that can be tiled with different perfect squares.

It isn’t difficult to do the tiling. All you have to remember is 32 × 33 and to put an 18 × 18 tile in a corner. The rest of the perfect square tiles seem to almost fall into place as this gif I made illustrates:

Tiling a 32 × 33 Rectangle

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Now I’ll tell you a little about the number 1056:
  • 1056 is a composite number.
  • Prime factorization: 1056 = 2 × 2 × 2 × 2 × 2 × 3 × 11, which can be written 1056 = 2⁵ × 3 × 11
  • 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) = 6 × 2 × 2 = 24. Therefore 1056 has exactly 24 factors.
  • Factors of 1056: 1, 2, 3, 4, 6, 8, 11, 12, 16, 22, 24, 32, 33, 44, 48, 66, 88, 96, 132, 176, 264, 352, 528, 1056
  • Factor pairs: 1056 = 1 × 1056, 2 × 528, 3 × 352, 4 × 264, 6 × 176, 8 × 132, 11 × 96, 12 × 88, 16 × 66, 22 × 48, 24 × 44, or 32 × 33
  • Taking the factor pair with the largest square number factor, we get √1056 = (√16)(√66) = 4√66 ≈ 32.49615
Since 1056 is the product of consecutive numbers, 32 × 33, it is the sum of the first 32 even numbers:
2 + 4 + 6 + 8 + 10 + . . .  +56 + 58 + 60 + 62 + 64 = 1056

 

How Can You Count These 1054 Tiny Squares?

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There are 1054 tiny squares in the image below, making 1054 a centered triangular number. How can you know that I’m not pulling the wool over your eyes about the number of tiny squares? Here are a few ways that you can quickly count all of them.

If you start with the yellow square in the center and count outward each succeeding triangle you will get 1 yellow square + 3 green squares + 6 blue squares + 9 purple squares + 12 red squares + 15 orange squares, etc. until you reach the final 78 blue squares:
1 + 3 + 6 + 9 + 12 + 15 + . . . + 78
= 1 + 3(1 + 2 + 3 + 4 + 5 + . . . + 26)
= 1 + 3(26*27)/2 = 1 + 3(351) = 1054

Using a little bit of algebra, you can show that
1 + 3(26*27)/2 = (3(26²) + 3(26) + 2)/2 = 1054

You can divide the centered triangle above into three triangles as I also did in the graphic. The three triangles represent the 25th, the 26th, and the 27th triangular numbers. Adding them up you get:
25(26)/2 + 26(27)/2 + 27(28)/2 = 325 + 351 + 378 = 1054

Here is some more information about the number 1054:

  • 1054 is a composite number.
  • Prime factorization: 1054 = 2 × 17 × 31
  • 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 1054 has exactly 8 factors.
  • Factors of 1054: 1, 2, 17, 31, 34, 62, 527, 1054
  • Factor pairs: 1054 = 1 × 1054, 2 × 527, 17 × 62, or 31 × 34
  • 1054 has no square factors that allow its square root to be simplified. √1054 ≈ 32.465366

1054 is the sum of six consecutive prime numbers:
163 + 167 + 173 + 179 + 181 + 191 = 1054

1054 is the hypotenuse of a Pythagorean triple:
496-930-1054 which is (8-15-17) times 62

1054 looks interesting when it is written in some other bases:
It’s 4A4 in BASE 15 (A is 10 base 10) because 4(15²) + 10(15) + 4(1) = 1054
1C1 in BASE 27 (C is 12 base 10) because 27² + 12(27) + 1 = 1054
VV in BASE 33 (V is 31 base 10) because 31(33) + 31(1) = 31(34) = 1054
V0 in BASE 34 because 31(34) = 1054

 

 

 

 

1051 is the 21st Centered Pentagonal Number

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1051 is the 21st centered pentagonal number. It is exactly 100 more than the previous centered pentagonal number because there are exactly 100 little blue squares on the outside-most pentagon in the graphic below.

Can you see the five triangles surrounding the center square? Each of them has the same number of tiny squares and indicates that 1051 is 1 more than five times the 20th triangular number:
1 + 5(20)(21)/2 = 1 + 50(21) = 1051

1049 and 1051 are twin primes.

  • 1051 is a prime number.
  • Prime factorization: 1051 is prime.
  • The exponent of prime number 1051 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1051 has exactly 2 factors.
  • Factors of 1051: 1, 1051
  • Factor pairs: 1051 = 1 × 1051
  • 1051 has no square factors that allow its square root to be simplified. √1051 ≈ 32.41913

How do we know that 1051 is a prime number? If 1051 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1051 ≈ 32.4. Since 1051 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1051 is a prime number.

1051 is a palindrome when it is written in three other bases:
It’s 737 in BASE 12 because 7(144) + 3(12) + 7(1) = 1051,
1H1 in BASE 25 (H is 17 base 10) because 25² +17(25) + 1 = 1051, and
151 in BASE 30 because 30² + 5(30) + 1 = 1051

1041 and Level 4

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Any level 3 puzzle can be easily made into a level 4 puzzle by removing some restrictions on the order of the clues. If you can solve a level 3 puzzle, then this level 4 puzzle will be only a little more difficult to solve.

Print the puzzles or type the solution in this excel file: 10-factors-1035-1043

What have I found out about the number 1041?

  • 1041 is a composite number.
  • Prime factorization: 1041 = 3 × 347
  • 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 1041 has exactly 4 factors.
  • Factors of 1041: 1, 3, 347, 1041
  • Factor pairs: 1041 = 1 × 1041 or 3 × 347
  • 1041 has no square factors that allow its square root to be simplified. √1041 ≈ 32.26453

1041 is also a palindrome in three other bases:
It’s 13131 in BASE 5 because 5⁴ + 3(5³) + 5² +3(5) + 1 = 1041,
545 in BASE 14 because 5(14²) + 4(14) + 5(1) = 1041, and
1E1 in BASE 26 (E is 14 base 10) because 26² + 14(26) + 1 = 1041

1035 is the 23rd Hexagonal Number

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1035 is the 23rd hexagonal number because of the way that it can be factored:
2(23²) – 23 = 1035,
(2(23) – 1)23 = 1035
or simply 45(23) = 1035.

Starting at the bottom of the hexagonal we see 1 yellow square, 5 green squares, 9 blue squares, 13 purple squares, 17 red squares, and 21 orange squares.

1, 5, 9, 13, 17, 21, . . . is an arithmetic progression or arithmetic sequence. The common difference between the numbers is 4.

The nth hexagonal number is the sum of the first n numbers in that arithmetic progression.
The first few hexagonal numbers form an arithmetic series: 1, 6, 15, 28, 45, 66 and so forth.
1035 is the sum of the first 23 numbers in the progression so it is the 23rd term in the series and the 23rd hexagonal number.

All hexagonal numbers are also triangular numbers. 1035 is the 45th triangular number because 45(46)/2 = 1035.

Starting in the lower left-hand corner of that triangle we see 1 yellow square, 2 green squares, 3 blue squares, 4 purple squares, 5 red squares, and 6 orange squares.

1, 2, 3, 4, 5, 6, . . .  is the simplest arithmetic progression there is. The common difference is 1.

The nth triangular number is the sum of the first n numbers in that arithmetic progression.

The first few triangular numbers form an arithmetic series: 1, 3,  6, 10, 15, 21, 28, 36, 45, 55, 66 and so forth.  (The blue triangular numbers are also hexagonal numbers.)
1035 is the sum of the first 45 numbers in the progression so it is the 45th term in the series and the 45th triangular number.

1035 is also the hypotenuse of one Pythagorean triple:
621-828-1035 which is (3-4-5) times 207

It is also a leg in several Pythagorean triples including
1035-1380-1725 which is (3-4-5) times 345

  • 1035 is a composite number.
  • Prime factorization: 1035 = 3 × 3 × 5 × 23, which can be written 1035 = 3² × 5 × 23
  • 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 = 23. Therefore 1035 has exactly 12 factors.
  • Factors of 1035: 1, 3, 5, 9, 15, 23, 45, 69, 115, 207, 345, 1035
  • Factor pairs: 1035 = 1 × 1035, 3 × 345, 5 × 207, 9 × 115, 15 × 69, or 23 × 45,
  • Taking the factor pair with the largest square number factor, we get √1035 = (√9)(√115) = 3√115 ≈ 32.1714

Level 2 and Simplifying √1014

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Today’s Puzzle:

Have you memorized a basic multiplication table? If you have, then you can solve this puzzle. The numbers being multiplied together aren’t where they are in a regular multiplication table, but you can still easily figure out where they need to go. There is only one solution. I bet you can find it!

Print the puzzles or type the solution in this excel file: 12 factors 1012-1018

Simplifying √1014:

If I wanted to find √1014, I would first check to see if it were divisible by 4 or by 9 because most numbers whose square roots can be simplified are divisible by 4 or by 9 or both.
1014 isn’t divisible by 4 because 14 isn’t divisible by 4.
It isn’t divisible by 9 because 1 + 0 + 1 + 4 = 6, and 6 is not divisible by 9.
However, it is divisible by both 2 and 3 and thus also by 6. Since most people are less likely to make a mistake dividing by 6 in ONE step instead of two, I would make a little division cake and do that division first:

Recognizing that 169 is a perfect square, I would then take the square root of everything on the outside of my little cake. (√6)(√169) = 13√6

Factors of 1014:

  • 1014 is a composite number.
  • Prime factorization: 1014 = 2 × 3 × 13 × 13, which can be written 1014 = 2 × 3 × 13²
  • The exponents in the prime factorization are 1, 1, and 2. Adding one to each and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 × 2 × 3 = 12. Therefore 1014 has exactly 12 factors.
  • Factors of 1014: 1, 2, 3, 6, 13, 26, 39, 78, 169, 338, 507, 1014
  • Factor pairs: 1014 = 1 × 1014, 2 × 507, 3 × 338, 6 × 169, 13 × 78, or 26 × 39,
  • Taking the factor pair with the largest square number factor, we get √1014 = (√169)(√6) = 13√6 ≈ 31.84337

Sum-Difference Puzzles:

6 has two factor pairs. One of those pairs adds up to 5, and the other one subtracts to 5. Put the factors in the appropriate boxes in the first puzzle.

1014 has six factor pairs. One of the factor pairs adds up to ­65, and a different one subtracts to 65. 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?

More about the Number 1014:

Because 13² is one of its factors, 1014 is the hypotenuse of two Pythagorean triples:
714-720-1014 which is 6 times (119-120-169),
390-936-1014 which is (5-12-13) times 78.

1014 looks interesting when written in some other bases:
It’s 600 in BASE 13 because 6(13²) = 6 (169) = 1014,
and 222 in BASE 22 because 2(22²) + 2(22) + 2(1) = 2(484 + 22 + 1) = 2(507) = 1014.

1009, a Prime Factor of the Year 2018

/ ivasallay / 8 Comments

Let’s begin with a mathematical equation you can use to countdown the final seconds of 2017 to welcome in the New Year, 2018.

2018 Equation

make misc GIFs like this at MakeaGif

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Okay, that equation had a lot of parentheses which made us multiply by 1 two different times. (Boring.) I designed it the way it is because I wanted to take advantage of the fact that 1009 × 2 = 2018.

This second equation created by Edmark M. Law needs no parentheses and is much more beautiful:

Countdown to 2018

make misc GIFs like this at MakeaGif

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Sometimes you might need a still picture instead of a gif.

This post will include lots of facts about 2018.

2018¹⁰⁻⁹⁻⁸⁺⁷⁺⁶⁻⁵⁻⁴⁺³⁺²⁻¹ = 2018

2018 can be written as the sum of four consecutive numbers:
503 + 504 + 505 + 506 = 2018

It can also be written as the sum of two consecutive even numbers:
1008 + 1010 = 2018

2018 will be an amazing year in many different areas:

That graphic is based on 43² +  13² = 2018.

This next one is based on the fact that 2(1009) = 2(28² +  15²) = 2018.

2018 is also the sum of three squares four different ways:

36² +  19² +  19² = 2018
35² +  28² +  3² = 2018
35² +  27² +  8² = 2018
33² +  23² +  20² = 2018

This next area problem is based on one of the sums of three squares listed above. Can you tell which one?

2018 is in exactly two Pythagorean triple triangles:
1118-1680-2018 and 2018-1018080-1018082.

In the triangle illustrated above, 2018 is about 500 times smaller than either of the other two sides. Yep, that graphic was definitely not drawn to scale.

By contrast, in this next triangle, hypotenuse 2018 is not even twice as big as either of the legs.

How did I find that triangle?
1118-1680-2018 can be calculated from 2(43)(13), 43² – 13², 43² + 13²
It is also 2 times (559-840-1009). That primitive triple can be calculated from
28² – 15², 2(28)(15), 28² + 15²

Is there any other relationship between 43² + 13² and 2(28² + 15²)? Yes.
28 + 15 = 43 and 28 – 15 = 13.

How did I find the triple with two sides in the millions (2018-1018080-1018082)?
2018 ÷ 2 = 1009 and 1009² ± 1 are the values of the other leg and the hypotenuse.

I like the way 2018 looks in these other bases:
It’s 8E8 in BASE 15 (E is 14 base 10) because 8(15²) + 14(15) + 8(1) = 2018,
2G2 in BASE 28 (16 is G base 10) because 2(28²) + 16(28) + 2(1) = 2018, and
2202202 in BASE 3 because 2(3⁶ + 3⁵ + 3³ + 3² +3⁰) = 2(1009) = 2018

This is how we can write 2018 as the sum of powers of 2:
2¹⁰ + 2⁹ + 2⁸ + 2⁷ + 2⁶ + 2⁵ + 2¹ = 2018

Finally, I give you my predictions of the factors we will see in 2018. You can be confident that these predictions will be 100% correct.
The positive factors for the year 2018 will be (drum roll) 1, 2, 1009, and 2018.
Sorry to say, but there will also be four negative factors of 2018: -1, -2, -1009, and -2018.

I also know that 2018 will have some complex factors because 43² +  13² = 2018.

Here is a graphic showing 2018’s factor pairs:

Related Articles:

  1. Edmark M. Law’s post titled Happy New Year 2018! (And Mathematical Facts about 2018) has many more mathematical curiosities about 2018.
  2. Mathwithbaddrawings.com humorously shares some upcoming mathematical dates and other facts about 2018 in Things to Know About the Year 2018.  At least one of those facts makes the number 2018 quite unique.
  3. 2018: Top Ten Facts about the New Year has a little bit of mathematics in it.

Since this is my 1009 post, I’ll tell you a few things about that number:

1009 is half of 2018.

1009 is the smallest four-digit prime number.

28² + 15² = 1009 so we get this Pythagorean triple:
559-840-1009

1009 is a palindrome or otherwise looks interesting in some other bases:
It’s 838 in BASE 11 because 8(121) + 3(11) + 8(1) = 1009,
474 in BASE 15 because 4(15²) + 7(15) + 4(1) = 1009,
321 in BASE 18 because 3(18²) + 2(18) + 1(1) = 1009,
2F2 in BASE 19 (F is 15 base 10) because 2(19²) + 15(19) + 2(1) = 1009,
1I1 in BASE 24 (I is 18 base 10) because 1(24²) + 18(24) + 1(1) = 1009, and
181 in BASE 28 because 1(28²) + 8(28) + 1(1) = 1009

  • 1009 is a prime number.
  • Prime factorization: 1009 is prime.
  • The exponent of prime number 1009 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1009 has exactly 2 factors.
  • Factors of 1009: 1, 1009
  • Factor pairs: 1009 = 1 × 1009
  • 1009 has no square factors that allow its square root to be simplified. √1009 ≈ 31.76476

How do we know that 1009 is a prime number? If 1009 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1009 ≈ 31.8. Since 1009 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1009 is a prime number.

Here’s another way we know that 1009 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 28² + 15² = 1009 with 28 and 15 having no common prime factors, 1009 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1009 ≈ 31.8. Since 1009 is not divisible by 5, 13, 17, or 29, we know that 1009 is a prime number.

Now I’ll include posts from twitter that celebrate 2018 mathematically. Some are easier to understand than others:

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js

https://platform.twitter.com/widgets.js
Be sure to click on this next one. There are MANY 2018 equations in the comments:

https://platform.twitter.com/widgets.js