Mathematics

168 and the Very Inspiring Blogger Award

/ ivasallay / 5 Comments

I Was Nominated for the Very Inspiring Blogger Award!

Nerdinthebrain is a very well-rounded and inspiring blogger. I feel quite honored that she nominated me for the Very Inspiring Blogger Award.

veryinspiring_bloggeraward

 

Just two little rules for accepting this award:

1) The nominee shall display the Very Inspiring Blogger Award logo on her/his blog, and link to the blog they got nominated from.

2) The nominee shall nominate fifteen (15) bloggers she/he admires, by linking to their blogs and informing them about it.

Because this award has these requirements, it may seem like a modern version of a chain letter, but it also appears to be a great way to step out of our comfort zones. It helps us read posts and share ideas with people with whom we have a little something in common but just don’t know it yet. Here are my 15 nominees:

  1. Beyondtraditionalmath
  2. Blogbloggerbloggest
  3. Crazygoodreaders DYSLEXIA DIGEST
  4. established1962
  5. Hummingtop
  6. colleenyoung.wordpress.com
  7. MY MATH-Y ADVENTURES
  8. Nebusresearch (He actually has 2 great blogs.)
  9. NumberLovingBeagle
  10. PeopleStoryNetwork
  11. Bookzoompa
  12. RobertLovesPi
  13. http://mathtuition88.com/
  14. VisuallyLiteral (Nancy Tordai Photography)
  15. MarekBennett

Congratulations to each of you. (I’ll do the informing tomorrow because I’ve already spent more time than usual on the internet today.)

Factors of 168:

√168 ≈ 12.96148. Let’s divide 168 by each number from 1 to 12 to find its factor pairs.

The prime factorization of 168 is 2³ × 3 × 7.
Adding 1 to each of the exponents in the prime factorization and multiplying, we get
(3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 12. Notice that 168 has exactly 12 factors.

More About the Number 168:

Four of those factor pairs are made up of only even numbers, so 168 is the difference of two squares four different ways:
43² – 41² = 168,
23² – 19² = 168,
17² – 11² = 168,
13² – 1² = 168.

Since 168 is 3 × 56, it is the sum of three consecutive numbers with 56 as the middle number:
55 + 56 + 57 = 168.

Since 168 is 7 × 24, it is the sum of seven consecutive numbers with 24 as the middle number:
21 + 22 + 23 + 24 + 25 + 26 + 27 = 168.

Finally, since 168 is divisible by 8, but not by 16, it is the sum of 16 consecutive numbers:
3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 = 168.

As a consequence of that last mathematical fact, here’s another way to make 168:
18² – 17² + 16² – 15² + 14² – 13² + 12² – 11² + 10² – 9² + 8² – 7² + 6² – 5² + 4² – 3² = 168.
I bet you weren’t expecting that!

168 = 6 × 28, so 168 is the product of the first two perfect numbers! Why are those numbers perfect? Each of them is the sum of their divisors:
6 = 1 + 2 + 3, and
28 = 1 + 2 + 4 + 7 + 14.

168 is a repdigit in several other bases:
It’s CC in base 13 because 12(13+1) = 168,
88 in base 20 because 8(20+1) = 168,
77 in base 23 because 7(23+1) = 168,
66 in base 27 because 6(27+1) = 168,
44 in base 41 because 4(41+1) = 168,
33 in base 55 because 3(55+1) = 168,
22 in base 83 because 2(83+1) = 168, and
11 in base 167 because 1(167+1) = 168.

 

151 and Why you shouldn’t even bother with that divisibility trick for 9 and 3

/ ivasallay / 2 Comments
  • 151 is a prime number.
  • Prime factorization: 151 is prime.
  • The exponent of prime number 151 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 151 has exactly 2 factors.
  • Factors of 151: 1, 151
  • Factor pairs: 151 = 1 x 151
  • 151 has no square factors that allow its square root to be simplified. √151 ≈ 12.2882

How do we know that 151 is a prime number? If 151 were not a prime number, then it would be divisible by at least one prime number less than or equal to √151 ≈ 12.3. Since 151 cannot be divided evenly by 2, 3, 5, 7, or 11, we know that 151 is a prime number.

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Don't add digits

I’m serious. Don’t bother using that trick that wants you to add up all the digits of a number to see if the number can be evenly divided by 3 or 9. Really, Don’t use it.

Even if it does take less time than actually dividing a whole number by 3 or by 9, it isn’t the best use of your time, and there are too many opportunities to make mistakes using it, especially if the whole number has more than a few digits.

One out of every three randomly chosen whole numbers will be divisible by 3. That’s pretty good odds if you guess, but I’m not advocating guessing.

I’m suggesting that you ignore most of the numbers and only add up a few them. Let me exaggerate the point I’m trying to make:  the following number is too big to put in your calculator, but would you really add up all its digits to see if it’s divisible by 3 or 9?

99999999999999999999997599999999999999999981999999999

No, you wouldn’t. You were probably able to tell that the number is divisible by 3 but not by 9 in seconds by ignoring most of the digits. What about this next number? It has the same digits but not in the same order. I’m sure you can tell almost as quickly that it is also divisible by 3 but not by 9.

99999999979999999999999599999999999999999991999999998

Yes, it’s true that the digits of most whole numbers aren’t mostly nines, but they can form small groups of numbers that are mostly multiples of nine. What are some common multiples of nine that you can ignore in any whole number? 0, 18, 27, 36, 45, 90, 234, 333, 567, 666, and even 1234567890. These digits can be ignored or eliminated no matter what order they appear in the whole number.

Let’s apply this process of elimination to a whole number that is too big for any standard calculator:

Process of Elimination

Look at the number at the top of this article, 473685. All of the digits can be eliminated so it is divisible by both 3 and 9 (but not by 6 because the original number wasn’t even.) See how quick and easy it can be to use this process of elimination?

So the next time you need to determine if a whole number is divisible by 3, 6, or 9, try ignoring or eliminating as many digits as you can before you do any addition. It will save you so much time. With some of that time you save you could complete a puzzle such as this one whose clues are all divisible by 3.

2014-24 Extra

 

140 and Gr-8 Divisibility Tricks

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140 is a composite number. Factor pairs: 140 = 1 x 140, 2 x 70, 4 x 35, 5 x 28, 7 x 20, 10 x 14. Factors of 140: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140. Prime factorization: 140 = 2 x 2 x 5 x 7, which can also be written 140 = 2² x 5 x 7.

140 is never a clue in the FIND THE FACTORS puzzles.

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In a previous post I discussed how to tell if a whole number can be evenly divided by 4, 25, or both just by looking at the last 2 digits of the number.

Mathematics is full of patterns so you might be wondering if there are divisibility tricks involving the last 3 digits of a number. Yes! There is!

divide by 125

Again, because we use base 10, and 2 x 5 = 10, the following divisibility tricks work:

  • 1000 (10 cubed) divides evenly into any number ending in 000.
  • 125 (5 x 5 x 5) divides evenly into any whole number that ends with 000, 125, 250, 375, 500, 625, 750, 875. [I remember all 8 endings by thinking about U.S. coins. (000) no quarters equals 00 cents, (125) 1 quarter equals 25 cents, (250) 2 quarters equals 50 cents, (375) 3 quarters equals 75 cents. Adding 500 to each of those endings will give us the rest.]
  • 8 (2 x 2 x 2) will divide evenly into whole numbers whose last 3 digits are divisible by 8.

Let’s explore that divisibility rule for eights a little more:

  • There are 125 different three digit endings that are divisible by 8. I will not list them here, but here is a trick to the trick.

A whole number whose 3rd to the last digit is EVEN is divisible by 8 if the last 2 digits are (00, 08, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96)

  • If you know the times tables up to 8 x 12, then you recognize ALL of those 2 digit numbers.
  • If the 3rd to the last digit is odd and the last 2 digits are divisible by 4 but not by 8 (04, 12, 20, 28, 36, 44, 52, 60, 68, 76, 84, 92), then the whole number is divisible by 8. This last rule can be expressed more concisely using those same 8’s multiplication facts listed above:

A whole number whose 3rd to the last digit is ODD is divisible by 8 if its last two digits ± 4 = (00, 08, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96)

I like playing with these tricks, and they save me time especially when it comes to factoring larger whole numbers. I hope you will enjoy playing with them as well!

138 and Divisibility Tricks 4 You

/ ivasallay / 6 Comments

138 is a composite number. Factor pairs: 138 = 1 x 138, 2 x 69, 3 x 46, or 6 x 23. Factors of 138: 1, 2, 3, 6, 23, 46, 69, 138. Prime factorization: 138 = 2 x 3 x 23.

138 is never a clue in the FIND THE FACTORS puzzles.

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After you learned some basic division facts, you probably realized:

  • 2 will divide evenly into any EVEN whole number.
  • 5 will divide evenly into whole numbers ending in 0 or 5.
  • 10 will divide evenly into whole numbers ending in 0.

These three rules are related to each other. All of them are true because we use base ten in our numbering system, and the prime factorization of 10 is 2 x 5.

If you needed to find the factors of a 33-digit whole number, you would be able to tell if 2, 5, or 10 divide evenly into it  just by looking at the last digit. 33-digits is more than a standard calculator can handle, but no matter how many digits a whole number has, as long as you can see the very last one, you can apply those three simple divisibility rules to know if 2, 5, or 10 are factors. Thus you will be able to do something a calculator can’t.

But wait, there are even more divisibility tricks if you can see the last TWO digits of the whole number!

divide by 4

  • 10 squared, better known as 100, divides evenly into any whole number ending in 00.
  • 5 x 5 = 25 which divides evenly into any whole number ending in 00, 25, 50, or 75.
  • 2^2 (AKA 4) divides evenly into a whole number if the final two digits can be divided evenly by 4.

How can one tell if the last two digits of a whole number are divisible by 4 (without actually dividing by 4)? I’ll show you how: I’ve put the 25 possible 2-digit multiples of 4 into one of two lists:

  • 00, 04, 08, 20, 24, 28, 40, 44, 48, 60, 64, 68, 80, 84, 88
  • 12, 16, 32, 36, 52, 56, 72, 76, 92, 96

Notice in the first list ALL the digits are even and the last digit (0, 4, or 8) can be divided evenly by 4.

Then look at the second list. The first digit is always odd and the last digit is either 2 or 6 (the only two even digits that are not divisible by 4).

Hmm. I think we can rewrite the divisibility rule for 4:

  • 4 (AKA 2^2) divides evenly into a whole number if the last two digits are even and the final digit is divisible by 4 (the last digit is 0, 4, or 8).
  • 4 divides evenly into any whole number whose next to the last digit is odd if the final digit is even but not divisible by 4 (the last digit is 2 or 6).

The rewritten divisibility rule is longer to read but takes a little less time to implement so you will have to decide which version of the rule works best for you. Either trick takes much less time than dividing some really long whole number by 4 or dividing by 2 twice.

Now I’m on to thinking about what the last THREE digits tell us.

91 and River crossings

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The age-old river crossing puzzle is humorously presented and beautifully illustrated with manipulatives that can easily be printed, cutout, and assembled. It will be a winner with children and adults alike.

The puzzle is this: The farmer needs to get a bag of corn, a chicken, and a fox across a river. The boat is small and can only hold the farmer and one other thing at a time. The farmer cannot ever leave the chicken with either the corn or the fox because one of them will be eaten before he returns. He can take several trips across the river, but how does he get all three across?

Pat’s Blog gives a history of the famous puzzle and its variations, and TV Tropes gives the solution to the puzzle here.

91 is a composite number. 91 = 1 x 91 or 7 x 13. Factors of 91: 1, 7, 13, 91. Prime factorization: 91 = 7 x 13.

91 is never a clue in the FIND THE FACTORS puzzles.

91 is prominent in this cool pattern:

 

78 and Patterns of Ideas (Hardy)

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78 is a composite number. 78 = 1 x 78, 2 x 39, 3 x 26, or 6 x 13. Factors of  78: 1, 2, 3, 6, 13, 26, 39, 78. Prime factorization: 78 = 2 x 3 x 13.

78 is never a clue in the FIND THE FACTORS 1-10 or 1-12 puzzles.

Here’s a sample of a very creative cartoon by Marek Bennett. I especially like this one because it features a quote from G. H. Hardy, a hero of mine. Because of Hardy, the world knows about Ramanujan, possibly the greatest mathematician of the 20th century.

66 and Fractal line

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66 is a composite number. 66 = 1 x 66, 2 x 33, 3 x 22, or 6 x 11. Factors of 66: 1, 2, 3, 6, 11, 22, 33, 66. Prime factorization: 66 = 2 x 3 x 11.

Sometimes 66 is a clue in the FIND THE FACTORS 1 – 12 puzzles. Even though it has other factors, the puzzle only uses 6 x 11 = 66.

I like this fractal animation very much! It really goes on forever!

52 and Maths in the snow

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Here are some incredible and beautiful sketches in the snow. These pictures must have been taken from an airplane or helicopter. Be sure to click on the picture so you can see many other sketches including a gorgeous fractal. I’m sure you want to know how this fabulous art was made.

52 is a composite number. 52 = 1 x 52, 2 x 26, or 4 x 13. Factors of 52: 1, 2, 4, 13, 26, 52. Prime factorization: 52 = 2² x 13.

52 is never a clue in the FIND THE FACTORS 1- 10 or 1- 12 puzzles.

51 and Magic forcing grid – Maths Magic

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Mathematics is full of magic, sometimes more than we even expected. There appears to be a connection between this math trick and 4 x 4 magic squares. I will be interested to see if anyone can form a 4 x 4 magic square from the information provided by The Science Magician in this post. There are 880 ways to make a 4 x 4 magic square, so it seems like it would still be a lot of work. This same process works on a 3 x 3 square and results in numbers that add up to 15 in every case when the numbers from 1 to 9 are used.

If you reverse the digits in 15, you get 51. Can you make a magic square in which the magic sum is 51? Yes, you can! If you use the numbers from 13 to 21, the magic sum will be 51. One such configuration is shown here:

Notice that 17 is the number in the middle square and 17 × 3 = 51.

Only multiples of 3 that are greater than or equal to 15 can be magic sums of a 3 × 3 magic square.

51 is a composite number. 51 = 1 x 51 or 3 x 17. Factors of 51: 1, 3, 17, 51. Prime factorization: 51 = 3 x 17.

51 is never a clue in the FIND THE FACTORS puzzles.

51 is in this curious pattern:

48 and What’s wrong with this?

/ ivasallay / 2 Comments

Here is a four piece puzzle arranged two different ways. There are 24 blue squares and 24 yellow squares. 24 + 24 = 48. Read on to see why the factors of 48 are significant to this puzzle.

Those triangles are some fabulous mathematical slight of hand.  Something is wrong with the puzzle, but what?

Something is fishy with these two triangle puzzles, too:

And with these two triangles made with six puzzle pieces!

If I had time, I could make an infinite number of these types of puzzles. Why do we get a solid triangle sometimes, but one with a hole in it the other times?

Believe it or not, the answer to the triangle puzzles has something to do with factoring.

Here’s that first puzzle again, but smaller:

The first triangle doesn’t have a hole it while the second one does because

Who would have thought that comparing fractions could turn into a type of magic trick?!!

Both triangles have 48 blue or yellow squares, but the second triangle wants you to think it should have 49.

(Psst….6/7 is also the slope of the green triangle and 7/8 is the slope of the red triangle. Since the two triangles don’t have identical slopes, the big “triangles” formed in the puzzles aren’t really triangles at all.)

I made all the puzzles in this post by comparing fractions. In each case, I simply used two consecutive composite numbers and then factored them and came up with the fractions I wanted to use. The numerators and the denominators of the fractions became the sides of the triangles.

Now, what do you think of this video of a never-ending chocolate bar?

Let me tell you a little bit about the number 48:

48 is a composite number. 48 = 1 x 48, 2 x 24, 3 x 16, 4 x 12, or 6 x 8. Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Prime factorization: 48 = 2 x 2 x 2 x 2 x 3, which can also be written 48 = 2⁴ x 3.

Sometimes 48 is a clue in the FIND THE FACTORS puzzles. Even though it has many other factors, we use only 6 x 8 for the FIND THE FACTORS 1-10 puzzles and 6 x 8 or 4 x 12 for the FIND THE FACTORS 1-12 puzzles.

 

factor tree