A Multiplication Based Logic Puzzle

Archive for the ‘Mathematics’ Category

1034 Find the Factor Challenge Puzzle

One of the things I really like about these Challenge puzzles is that you have to use logic. Guessing and checking just won’t work. Go ahead and give this puzzle a try!

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

1 – 0 + 3 – 4 = 0, so 1034 can be evenly divided by 11.

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Level 2 and Simplifying √1014

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

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

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Here are some facts 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

  • 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

1009, a Prime Factor of the Year 2018

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

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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

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

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

1005 and Level 4

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

996 Christmas Factor Tree

The puzzles this week might look rather plain, but together the seven puzzles make a lovely Christmas tree factoring puzzle. The difficulty level of each of the puzzles is not identified. Some of them are very easy, and some of them are difficult. Some are in-between. How many of them can you solve?

Print the puzzles or type the solution in this excel file: 12 factors 993-1001

Here’s a little about the number 996:

Here are a few of its possible factor trees. They look a little like Christmas trees, too.

 

Usually,  I only go up to base 36 when I look for palindromes or repdigits. 966 is NOT a palindrome or repdigit in any of those bases, but can it ever be one? To me, repdigits are more interesting than palindromes because you can find them by factoring.  966 has 6 factors greater than 36: 83, 166, 249, 332, 498, 996. If you subtract 1 from each of those, then 996 will be a repdigit in each of those bases.
In BASE 82, it’s CC (C is 12 base 10)
In BASE 165, it’s 66
In BASE 248, it’s 44
In BASE 331, it’s 33
In BASE 497, it’s 22
In BASE 995, it’s 11
Don’t be surprised when I tell you that 12, 6, 4, 3, 2, and 1 are also factors of 996!

  • 996 is a composite number.
  • Prime factorization: 996 = 2 × 2 × 3 × 83, which can be written 996 = 2² × 3 × 83
  • 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 996 has exactly 12 factors.
  • Factors of 996: 1, 2, 3, 4, 6, 12, 83, 166, 249, 332, 498, 996
  • Factor pairs: 996 = 1 × 996, 2 × 498, 3 × 332, 4 × 249, 6 × 166, or 12 × 83,
  • Taking the factor pair with the largest square number factor, we get √996 = (√4)(√249) = 2√249 ≈ 31.55947

985 Make an Origami Santa Star

Almost immediately when I saw Paula Beardell Krieg’s origami stars, I thought about turning one into a Santa Star. It occurred to me that the white pentagon formed on the back of the star would make a nice beard for Santa. I made a prototype and tweaked it and tweaked it until I got this result:

Why did I want to do this? My daughter-in-law, Michelle, adores Santa Stars. They are her favorite Christmas decoration. When I gave her this Santa Star, she got so excited. She recently took a picture of her collection, and I am thrilled that the one I made for her was included.

If you would like to make this Santa Star, follow these steps.

  1. Click on the pentagon above, then copy and paste it into a document. Make it as big as your printer allows.
  2. Print the pentagon and cut it out.
  3. Follow the directions in the video by Tobias that Paula recommends.

Here are some pictures I took as I folded mine. Click on them if you want to see them better. I’ve also included a few tips to help you in folding the star:

In this picture, you can see that the pentagon was folded in half five different ways in the first set of folds. The second set of folds creates a smaller pentagon in the center of the pentagon as well as a star-like shape.

The third set of folds creates a new crease. I make a flower-like shape by refolding that crease on each side. To me, this “flower” is a very important step to get the paper to form the star.

Those creases will help form the small white pentagon you see in the picture below that will become Santa’s beard.

Turn the paper over to reveal a bigger pentagon. You will fold the vertices of this pentagon to the center of the pentagon. Fold the red and black tips at the same time as you fold the vertices. After you make the first fold, I recommend unfolding it. Fold the other vertices in order so that first fold will eventually become your last fold. The last fold is the most difficult to do. If it has already been folded once, it will be much easier to fold at the end.

Again, here is the finished Santa star.

Now I’ll share some facts about the number 985:

985 is the sum of three consecutive prime numbers:
317 + 331 + 337 = 985

29² + 12² = 985 and 27² + 16² = 985

985 is the hypotenuse of FOUR Pythagorean triples
140-975-985
591-788-985
473-864-985
696-697-985

When is 985 a palindrome?
It’s 505 in BASE 14 because 5(14²) + 5(1) = 5(196 + 1) = 5(197) = 985
It’s 1H1 BASE in 24 (H is 17 base 10) because 1(24²) + 17(24) + 1(1) = 985

  • 985 is a composite number.
  • Prime factorization: 985 = 5 × 197
  • 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 985 has exactly 4 factors.
  • Factors of 985: 1, 5, 197, 985
  • Factor pairs: 985 = 1 × 985 or 5 × 197
  • 985 has no square factors that allow its square root to be simplified. √985 ≈ 31.3847

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