Month: March 2015

433 The Whole Is Greater Than the Sum of Its Parts

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Today I practiced a song with a choir comprised of nine smaller choirs. Every time before I had only practiced the song with our smaller choir. As I listened to all the voices in this big choir while we sang in harmony, I thought, “A big choir sounds SO good. The whole is greater than the sum of its parts!”

The Whole Is Greater Than The Sum of Its Parts - Aristotle

Here is a professional recording of the beautiful song we are singing. The music in this youtube video is directed by the composer, Rob Gardner. It features the London Symphony Orchestra at Air Studios, London and the Spire Choir at Camelback Baptist Church in Paradise Valley, Arizona.

The number 433 is also greater than the sum of its parts.

12² + 17² = 433

A Chinese checker board looks like a star and has 121 holes so 121 is called a star number. The first star number is 1, followed by 13, 37, 73, and 121. The ninth star number is 433.

433 is one more than twelve times the eighth triangular number. Star numbers and triangular numbers are related like that.

145-408-433 is a primitive Pythagorean triple.

Here is the factoring information for the number 433:

  • 433 is a prime number. 431 and 433 are twin primes.
  • Prime factorization: 433 is prime.
  • The exponent of prime number 433 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 433 has exactly 2 factors.
  • Factors of 433: 1, 433
  • Factor pairs: 433 = 1 x 433
  • 433 has no square factors that allow its square root to be simplified. √433 ≈ 20.8087

How do we know that 433 is a prime number? If 433 were not a prime number, then it would be divisible by at least one prime number less than or equal to √433 ≈ 20.8087. Since 433 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 433 is a prime number.

 

How to Simplify √432

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432 factor tree

Many people use factor trees and prime factorizations to figure out how to simplify square roots. I don’t. I don’t see the point in breaking something completely apart just to put it back together again, especially when it ISN’T necessary to break it completely apart.

So instead of using a factor tree, I’ve modified the cake method, and I use it to simplify square roots.

For example, to find the square root of 500, I would never use its prime factorization: 500 = 2 x 2 x 5 x 5 x 5. Instead, I would first divide 500 by perfect square 100 to get 5. Then I would take the square root of both 100 and 5 to get 10√5.

Only 1% of numbers are divisible by 100, but for the ones that are, I always start by dividing by 100.

Now get this: Roughly 82.5% of all numbers that have simplifiable square roots can be evenly divided by perfect squares four and/or by nine. It is so easy to tell if a number can be evenly divided by either of those numbers. That is why I always start with one hundred, then four, then nine.

Let’s look at 432: Since the number formed from the last two digits, 32, is divisible by 4, I know that 432 is also divisible by four, so I will go ahead and do the division.

432 divided by 4

Now I look at 108. The last two digits, 08, can also be evenly divided by 4, so I do that division as well:

108 divided by 4

108 divided by 4 gives us 27 which is not divisible by 4. However since 2 + 7 = 9, I know that 27 is divisible by 9, so I do that division next:

27 divided by 9

27 divided by 9 is 3. Since the only square number that will divide evenly into 3 is 1, I’m done with the division process. Now I take the square roots of all the numbers on the outside of the cake and multiply them together: √(4 x 4) x (√9) x (√3) = 4 x 3 x (√3) = 12√3.

I will give other examples of this method in future posts. Here’s today’s puzzle:

432 Puzzle

Print the puzzles or type the factors on this excel file:10 Factors 2015-03-16

  • 432 is a composite number.
  • Prime factorization: 432 = 2 x 2 x 2 x 2 x 3 x 3 x 3, which can be written 432 = (2^4) x (3^3)
  • The exponents in the prime factorization are 4 and 3. Adding one to each and multiplying we get (4 + 1)(3 + 1) = 5 x 4 = 20. Therefore 432 has exactly 20 factors.
  • Factors of 432: 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 36, 48, 54, 72, 108, 144, 216, 432
  • Factor pairs: 432 = 1 x 432, 2 x 216, 3 x 144, 4 x 108, 6 x 72, 8 x 54, 9 x 48, 12 x 36, 16 x 27, or 18 x 24
  • Taking the factor pair with the largest square number factor, we get √432 = (√144)(√3) = 12√3 ≈ 20.7846

432 Logic

431 and Level 5

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431 is the sum of seven consecutive prime numbers beginning with prime number 47. What other prime numbers are in that sum? If you would like, you can type your answer in the comments.

2^431 (mod 431) = 2 so 431 is VERY likely a prime number. Scroll down past the puzzle to know for sure.

431 Puzzle

Print the puzzles or type the factors on this excel file:10 Factors 2015-03-16

  • 431 is a prime number.
  • Prime factorization: 431 is prime.
  • The exponent of prime number 431 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 431 has exactly 2 factors.
  • Factors of 431: 1, 431
  • Factor pairs: 431 = 1 x 431
  • 431 has no square factors that allow its square root to be simplified. √431 ≈ 20.7605

How do we know that 431 is a prime number? If 431 were not a prime number, then it would be divisible by at least one prime number less than or equal to √431 ≈ 20.7605. Since 431 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 431 is a prime number.

431 Logic

430 and Level 4

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The factors of 430 are listed after the puzzle.

What interesting fact could I say about the number 430? I did a little research.

Biblewheel.com quotes a verse from Galatians 3:17 in the Bible: “And this I say, that the covenant, that was confirmed before of God in Christ, the law, which was four hundred and thirty years after, cannot disannul, that it should make the promise of none effect.” Biblewheel states that 430 means the law, just as 666 means the beast.

OEIS.org says that 430 “is the number of necklaces possible with 6 beads, each being one of 4 colors.” If I understood what that means in English, I’m sure I could construct it mathematically. I know it can’t mean 6 Choose 4. If you understand what it means, please comment.

430 Puzzle

Print the puzzles or type the factors on this excel file:10 Factors 2015-03-16

  • 430 is a composite number.
  • Prime factorization: 430 = 2 x 5 x 43
  • 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 x 2 x 2 = 8. Therefore 430 has exactly 8 factors.
  • Factors of 430: 1, 2, 5, 10, 43, 86, 215, 430
  • Factor pairs: 430 = 1 x 430, 2 x 215, 5 x 86, or 10 x 43
  • 430 has no square factors that allow its square root to be simplified. √430 ≈ 20.7364

430 Logic

Level 3 and Divisibility Tricks Applied to 429

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Today my blog has had 42 referral views from twitter, mostly thanks to other people’s tweets. Thank you for the tweets and retweets. Also, welcome everyone who visits my blog because of them or for any other reason!

429 Puzzle

Print the puzzles or type the factors on this excel file:10 Factors 2015-03-16

Let’s try some divisibility tricks on the number 429:

4 + 2 is 6, a multiple of three, so 429 is a divisible by three. Why didn’t I include 9 in the sum? Because I already know that 9 is a multiple of three, so it is redundant to include it!

Now try another divisibility trick on 429: I’ve put the odd numbered digits in bold and the even numbered digits in regular type. If we sum the odd numbered digits and subtract the even numbered digit, we get 4 + 9 – 2 = 11, a multiple of 11. That means that 429 is also divisible by 11!

  • 429 is a composite number.
  • Prime factorization: 429 = 3 x 11 x 13
  • 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 x 2 x 2 = 8. Therefore 429 has exactly 8 factors.
  • Factors of 429: 1, 3, 11, 13, 33, 39, 143, 429
  • Factor pairs: 429 = 1 x 429, 3 x 143, 11 x 39, or 13 x 33
  • 429 has no square factors that allow its square root to be simplified. √429 ≈ 20.7123

A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. Write the corresponding factors in the factor column (1st column) and factor row (top row).  Because this is a level three puzzle, you have now written a factor at the top of the factor column. Continue to work from the top of the factor column to the bottom, finding factors and filling in the factor column and the factor row one cell at a time as you go.

429 Factors

428 St. Patrick’s Day Smile

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Happy Saint Patrick’s Day! Today’s puzzle reminds me of a smile, and I hope it brightens your day.

428 is divisible by 4 because 28 is divisible by 4.

Since 28 can be evenly divided by 4, but not by 8, we know that ANY number ending in 128, 328, 528, 728, and 928 will also be divisible by 8. Notice all those odd digits just before the 28? Likewise, we know that ANY number ending in 028, 228, 428, 628, or 828 will NOT be divisible by 8.

All of the factors of 428 are listed below today’s  puzzle.

428 Puzzle

Print the puzzles or type the factors on this excel file:10 Factors 2015-03-16

  • 428 is a composite number.
  • Prime factorization: 428 = 2 x 2 x 107, which can be written 428 = (2^2) x 107
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 428 has exactly 6 factors.
  • Factors of 428: 1, 2, 4, 107, 214, 428
  • Factor pairs: 428 = 1 x 428, 2 x 214, or 4 x 107
  • Taking the factor pair with the largest square number factor, we get √428 = (√4)(√107) = 2√107 ≈ 20.6882

428 Factors

427 and Level 1

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Because one of 427’s factors is the hypotenuse of a primitive Pythagorean triple, 427 is also the hypotenuse of the Pythagorean triple, 77-420-427.

What was that primitive Pythagorean triple? Scroll down past the puzzle to see if you were right.

427 Puzzle

Print the puzzles or type the factors on this excel file:10 Factors 2015-03-16

[77-420-427] is [11-60-61] times seven.

  • 427 is a composite number.
  • Prime factorization: 427 = 7 x 61
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 x 2 = 4. Therefore 427 has exactly 4 factors.
  • Factors of 427: 1, 7, 61, 427
  • Factor pairs: 427 = 1 x 427 or 7 x 61
  • 427 has no square factors that allow its square root to be simplified. √427 ≈ 20.6640

427 Factors

426 My Response to a Pi-lish Question

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Hungarian Pi

A comma is used for decimals in many countries.

This last week there was a post on the Mathemagical Site titled “Do You Speak Pilish?”  Some people remember the digits of π by memorizing carefully constructed sentences in which the first word has three letters, the second word has one letter, and so on. Several examples were given, not just in English, but in eight other languages as well!

Hungarian was not one of the languages listed, but I wondered if there could possibly be a Pilish way for Hungarians to remember the digits of pi? (Almost all of my husband’s relatives were born in Hungary, and I am fascinated with the country and the language.) I just had to google “Magyar pi szám,” to find an article titled Minden idők legjobb magyar nyelvű pi-verse.

Now while I can read many Hungarian words, the sentence structure is so different from English that my comprehension isn’t as good as I’d like it to be. My son, David, taught himself the basics of the language before he went there to live and work several years ago. I emailed him the article requesting that he help me with the translation. In the email he sent back you will notice the problem with word for word translation of Hungarian into English. My son wrote:

“I don’t think I could translate it whilst maintaining the word lengths (which is the whole point). I’m giving it to you with a more or less word for word translation along with one that is written in more natural English. The Ludolph it mentions in the poem is the Dutch mathematician Ludolph van Ceulen, who was the first to publish pi up to 20 digits.”

I put his word for word translation in the following graphic:

Hungarian Pilish Pi

 

Here is David’s translation into more natural English:

  • Instead of the old and rough approximation,
  • Count the letters that come, word for word
  • If we end here at twenty words, we already have Ludolph’s result,
  • but exactly 10 more come from this last stanza.
  • That, I can promise confidently.”

Here is my answer to the question, “Do you speak Pilish?”

Not really. I am not the least bit interested in memorizing some cute paragraph in English to help me remember the first 30 or so digits of pi, BUT in Hungarian, I am going to give a try!

  • 426 is a composite number.
  • Prime factorization: 426 = 2 x 3 x 71
  • 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 x 2 x 2 = 8. Therefore 426 has exactly 8 factors.
  • Factors of 426: 1, 2, 3, 6, 71, 142, 213, 426
  • Factor pairs: 426 = 1 x 426, 2 x 213, 3 x 142, or 6 x 71
  • 426 has no square factors that allow its square root to be simplified. √426 ≈ 20.6398

425 and Level 6

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425 ends in 25 so it can be divided evenly by 25. If I had $4.25 all in quarters. How many quarters would I have? That’s the problem that I think of when I divide by 25. All of the factors of 425 are listed below the puzzle.

For some reason unknown to me, here in the United States, dates are ordered by month, date, and year. This rather illogical way of ordering allows us to say that today is 3-14-15, which are the first five digits of pi.  It could also be said that 3-14-15 at 9:26:53 gives the first ten digits of pi.

Logical or not, it is fun to declare today as Pi Day. Today’s puzzle celebrates those first five digits:

425 Puzzle

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-09

  • 425 is a composite number.
  • Prime factorization: 425 = 5 x 5 x 17, which can be written 425 = (5^2) x 17
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 x 2  = 6. Therefore 425 has exactly 6 factors.
  • Factors of 425: 1, 5, 17, 25, 85, 425
  • Factor pairs: 425 = 1 x 425, 5 x 85, or 17 x 25
  • Taking the factor pair with the largest square number factor, we get √425 = (√25)(√17) = 5√17 ≈ 20.6155

425 and all of it factors (except 1) are hypotenuses of primitive Pythagorean triples, so 425 is the hypotenuse of several triples:

  • [87-416-425] and
  • [297-304-425] are primitives
  • [65-420-425] is [13-84-85] times 5
  • [119-408-425] is [7-24-25] times 17
  • [180-385-425] is [36-77-85] times 5
  • [200-375-425] is [8-15-17] times 25
  • [255-340-425] is [3-4-5] times 85

425 Logic

424 and Level 5

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The last two digits of 424 can be evenly divided by 4 so 424 can also be evenly divided by 4. Also because 24 is divisible by 8 and because the digit preceding the 24 is even, 424 is also divisible by 8. All of the factors of 424 are listed below the puzzle.

424 Puzzle

Print the puzzles or type the factors on this excel file: 12 Factors 2015-03-09

  • 424 is a composite number.
  • Prime factorization: 424 = 2 x 2 x 2 x 53, which can be written 424 = (2^3) x 53
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 424 has exactly 8 factors.
  • Factors of 424: 1, 2, 4, 8, 53, 106, 212, 424
  • Factor pairs: 424 = 1 x 424, 2 x 212, 4 x 106, or 8 x 53
  • Taking the factor pair with the largest square number factor, we get √424 = (√4)(√106) = 2√106 ≈ 20.5913

424 Logic