A Multiplication Based Logic Puzzle

Thank you, dear readers, for stopping by today and/or any other time you have visited! My blog had the best year ever in 2017, and it is all because of you. I sincerely hope that I have been of service to you.

Here’s to you, dear friend. May lifting each other help us both to climb even higher this coming year!

May you be able to find the factors in this puzzle and, more importantly, the factors in life that will bring you happiness and success!

Print the puzzles or type the solution in this excel file: 10-factors-1002-1011

Let me tell you some things about the number 1011 that you probably didn’t know before:

The only nonzero digits in 1011 are three 1’s so 1011 is divisible by 3, and 1011 is included in this interesting pattern:

1011² = 1022121; Notice that the digits in bold are 1011 and all the other digits are 2’s. Thank you Stetson.edu for that fun fact.

1011 looks interesting in a few other bases:
It’s 33303 in BASE 4 because 3(4⁴ + 4³ + 4² + 4⁰) = 3(256 + 64 + 16 + 1) = 3(337) = 1011,
323 in BASE 18 because 3(18²) + 2(18) + 3(1) = 1011

1011 is the hypotenuse of a Pythagorean triple:
525-864-1011 which is 3 times (175-288-337)

  • 1011 is a composite number.
  • Prime factorization: 1011 = 3 × 337
  • 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 1011 has exactly 4 factors.
  • Factors of 1011: 1, 3, 337, 1011
  • Factor pairs: 1011 = 1 × 1011 or 3 × 337
  • 1011 has no square factors that allow its square root to be simplified. √1011 ≈ 31.7962

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Level 6 puzzles are difficult. At least they are until you’ve done a few, then they become much less difficult to solve.

This Find the Factors 1 – 10 Challenge is much more difficult. I published the first of these puzzles (#1000) last week, but I won’t necessarily make one every week. If you’ve done a few level 6 puzzles, see how you do with this one.

Print the puzzles or type the solution in this excel file: 10-factors-1002-1011

Here’s a little about the number 1010:

1010 is the sum of two squares two different ways:
29² + 13² = 1010
31² + 7² = 1010

That means that 1010 is the hypotenuse of some Pythagorean triples:
200-990-1010 which is 10 times (20-99-101)
434-912-1010 calculated from 2(31)(7), 31² – 7², 31² + 7²
606-808-1010 which is (3-4-5) times 202
672-754-1010 calculated from 29² – 13², 2(29)(13), 29² + 13²

1010 is a fun-looking number in base 10.
It is also palindrome 262 in BASE 21 because 2(21²) + 6(21) + 2(1) = 1010

  • 1010 is a composite number.
  • Prime factorization: 1010 = 2 × 5 × 101
  • 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 1010 has exactly 8 factors.
  • Factors of 1010: 1, 2, 5, 10, 101, 202, 505, 1010
  • Factor pairs: 1010 = 1 × 1010, 2 × 505, 5 × 202, or 10 × 101
  • 1010 has no square factors that allow its square root to be simplified. √1010 ≈ 31.780497

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

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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

The number 1008 has so many factors that I just had to make a forest of some of its MANY possible factor trees. 1008 has fifteen factor pairs. I’ve made a factor tree for every factor pair, except 1 × 1008. I’ll start off with these short, wide, beautiful trees that feature six of 1008’s factor pairs:

Notice that no matter what factor pair we use, each tree has the same prime factors that I have highlighted in red. Even this lean, thin very-easy-to-read tree uses those same prime factors:

Finally, here are eight more factor trees that begin with the other eight factor pairs for 1008. They aren’t as good-looking as all the trees above, but they still work as factor trees and help us find those same red prime factors for 1008:

Here are some other facts about the number 1008:

1008 is the sum of ten consecutive prime numbers:
79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 = 1008

1008 looks interesting when written in some other bases:
It’s 33300 in BASE 4 because 3(4⁴) + 3(4³) + 3(4²) = 3(256 + 64 + 16) = 3(336) = 1008,
4400 in BASE 6 because 4(6³) + 4(6²) = 4(216 + 36) = 4(252) = 1008,
700 in BASE 12 because 7(12²) = 7(144) = 1008
SS in BASE 35 (S is 28 base 10) because 28(35) + 28(1) = 28(36) = 1008
S0 in BASE 36 because 28(36) = 1008

  • 1008 is a composite number.
  • Prime factorization: 1008 = 2 × 2 × 2 × 2 × 3 × 3 × 7, which can be written 1008 = 2⁴ × 3² × 7
  • The exponents in the prime factorization are 4, 2 and 1. Adding one to each and multiplying we get (4 + 1)(2 + 1)(1 + 1) = 5 × 3 × 2 = 30. Therefore 1008 has exactly 30 factors.
  • Factors of 1008: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 28, 36, 42, 48, 56, 63, 72, 84, 112, 126, 144, 168, 252, 336, 504, 1008
  • Factor pairs: 1008 = 1 × 1008, 2 × 504, 3 × 336, 4 × 252, 6 × 168, 7 × 144, 8 × 126, 9 × 112, 12 × 84, 14 × 72, 16 × 63, 18 × 56, 21 × 48, 24 × 42, or 28 × 36
  • Taking the factor pair with the largest square number factor, we get √1008 = (√144)(√7) = 12√7 ≈ 31.74902

Some of the clues in this puzzle pair up in the same column or the same row and try to trick you into picking the wrong common factor. Nevertheless, the 10 clues in the puzzle work together to give you the most logical place to start the puzzle. It may be a little difficult to see the logic for this one but stick with it. You’ll figure it out.

Print the puzzles or type the solution in this excel file: 10-factors-1002-1011

Here’s a little bit about the number 1007:

1007 is the hypotenuse of a Pythagorean triple:
532-855-1007lwhich is 19 times (28-45-53)

1007 is palindrome 33233 in BASE 4
because 3(4⁴) + 3(4³) + 2(4²) + 3(4¹) + 3(4⁰) = 1007

  • 1007 is a composite number.
  • Prime factorization: 1007 = 19 × 53
  • 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 1007 has exactly 4 factors.
  • Factors of 1007: 1, 19, 53, 1007
  • Factor pairs: 1007 = 1 × 1007 or 19 × 53
  • 1007 has no square factors that allow its square root to be simplified. √1007 ≈ 31.73326

1006 and Level 5

Just like when I solved the previous puzzle, several times I had to pick a factor and see if there was only one place in the first column or the top row where it could go. Find as many of the other factors as you can before you employ that strategy, but when you need to use it, go for it. It is often very helpful!

 

Print the puzzles or type the solution in this excel file: 10-factors-1002-1011

At first, this puzzle is fairly easy to solve, but before long you will probably get stuck. To get unstuck, pick a number. See if there is only one place in the first column or the top row where that number can go. I had to use that strategy over and over again to solve this particular puzzle. Good luck!

Print the puzzles or type the solution in this excel file: 10-factors-1002-1011

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