427 and Level 1

/ ivasallay / Leave a comment

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

/ ivasallay / 3 Comments
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

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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

423 and Level 4

/ ivasallay / 2 Comments

4 + 2 + 3 = 9 therefore 423 can be evenly divided by both 3 and 9. Thus we know right away that it is a composite number whose square root can be simplified.

Also ANY natural number ending in 24 or 25 has a reducible square root, so with 423, 424, and 425 we have THREE consecutive numbers with reducible square roots. Sadly, 426 breaks the pattern so we do not have a fourth.

423 Puzzle

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

  • 423 is a composite number.
  • Prime factorization: 423 = 3 x 3 x 47, which can be written 423 = (3^2) x 47
  • 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 423 has exactly 6 factors.
  • Factors of 423: 1, 3, 9, 47, 141, 423
  • Factor pairs: 423 = 1 x 423, 3 x 141, or 9 x 47
  • Taking the factor pair with the largest square number factor, we get √423 = (√9)(√47) = 3√47 ≈ 20.5670

423 Logic

422 and Level 3

/ ivasallay / 6 Comments

422 has three even digits. How many factors could it possibly have? Scroll down to see.

This Find the Factors puzzle has the same color scheme that puzzles 414 and 417 had, but this one also has a couple of numbers that aren’t colored at all.  If 422 were colored, it would be yellow. I’ve had one very close guess posted in the comments for 417. Will anybody be able to read my mind and figure out what the coloring is all about? You can type your guess in the comments. Only an elementary education is required to figure it out!

422 Puzzle

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

  • 422 is a composite number.
  • Prime factorization: 422 = 2 x 211
  • 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 422 has exactly 4 factors.
  • Factors of 422: 1, 2, 211, 422
  • Factor pairs: 422 = 1 x 422 or 2 x 211
  • 422 has no square factors that allow its square root to be simplified. √422 ≈ 20.5426

A Logical Approach to FIND THE FACTORS: 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.

422 Factors

421 and Level 2

/ ivasallay / Leave a comment

We can easily see that 421 cannot be evenly divided by 2, 3, or 5. Could it possibly be a prime number? Scroll down to know for sure.

421 Puzzle

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

  • 421 is a prime number. 419 and 421 are twin primes.
  • Prime factorization: 421 is prime.
  • The exponent of prime number 421 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 421 has exactly 2 factors.
  • Factors of 421: 1, 421
  • Factor pairs: 421 = 1 x 421
  • 421 has no square factors that allow its square root to be simplified. √421 ≈ 20.5183

How do we know that 421 is a prime number? If 421 were not a prime number, then it would be divisible by at least one prime number less than or equal to √421 ≈ 20.5183. Since 421 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 421 is a prime number.

421 Factors

420 Factor Trees

/ ivasallay / 2 Comments

Did you know that the sum of all the prime numbers between 100 and 110 equals 420? Yes, 101 + 103 + 107 + 109 = 420.

Since 20 × 21 = 420, we know that 420 is the sum of the first 20 EVEN numbers. Thus,

  • 2 + 4 + 6 + 8 + 10 + 12 + 14 + 16 + 18 + 20 + 22 + 24 + 26 + 28 + 30 + 32 + 34 + 36 + 38 + 40 = 420.

420 is the smallest number that can be divided evenly by all the natural numbers from 1 to 7.

420 has a LOT of factors, more than most people would think it does.  In fact, of all of the numbers from 1 to 420, there is only one number, 360, that has as many factors as 420 has.

There are 4 different prime numbers that can divide evenly into 420. Here are those factor trees:

420 prime number factor trees

Many, but not all, of the factors of 420 are listed somewhere on those four trees.

  • 420 is a composite number.
  • Prime factorization: 420 = 2 x 2 x 3 x 5 x 7, which can be written 420 = (2^2) x 3 x 5 x 7
  • The exponents in the prime factorization are 2, 1, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 x 2 = 24. Therefore 420 has exactly 24 factors.
  • Factors of 420: 1, 2, 3, 4, 5, 6, 7, 10, 12, 14, 15, 20, 21, 28, 30, 35, 42, 60, 70, 84, 105, 140, 210, 420
  • Factor pairs: 420 = 1 x 420, 2 x 210, 3 x 140, 4 x 105, 5 x 84, 6 x 70, 7 x 60, 10 x 42, 12 x 35, 14 x 30, 15 x 28, or 20 x 21
  • Taking the factor pair with the largest square number factor, we get √420 = (√4)(√105) = 2√105 ≈ 20.4939

Each factor pair, except 1 x 420, can make its own factor tree. Here are some factor trees featuring the other seven factor pairs:

420 other factor trees

If I hadn’t made all the prime numbers in red, it’s possible that one or more of the prime numbers might get forgotten. That is why I prefer the cake method for finding the prime factorization of a number. All of the prime numbers are listed in numerical order on the outside of the cake.

Finding prime factors of 420

 

 

419 and Level 1

/ ivasallay / 4 Comments

2^419 (mod 419) = 2 so 419 is VERY LIKELY a prime number. Scroll down past the puzzle to know for sure.

419 Puzzle

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

  • 419 is a prime number.
  • Prime factorization: 419 is prime.
  • The exponent of prime number 419 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 419 has exactly 2 factors.
  • Factors of 419: 1, 419
  • Factor pairs: 419 = 1 x 419
  • 419 has no square factors that allow its square root to be simplified. √419 ≈ 20.4695

How do we know that 419 is a prime number? If 419 were not a prime number, then it would be divisible by at least one prime number less than or equal to √419 ≈ 20.4695. Since 419 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 419 is a prime number.

419 Factors

A Teacher Affects Eternity and Divisibility Tricks Applied to 418

/ ivasallay / Leave a comment

Henry Brooks Adams said, “A teacher affects eternity; he can never tell where his influence stops.” Of course the quote applies to female teachers just as much as it does to male teachers. Affecting eternity is pretty serious business.

It will probably take less than two minutes to read Marek Bennet’s  comic on how the work you do fits into the educational system. Its message is worth pondering for quite a long time.

Let me demonstrate some quick prime number divisibility tricks on the number 418:

  • It’s even so its divisible by 2.
  • 4 + 1 + 8 = 13, which is not a multiple of 3, so 418 is not divisible by 3.
  • The last digit isn’t 0 or 5, so 418 is not divisible by 5.
  • 41 – 2(8) = 41 – 16 = 25, which is not a multiple of 7, so 418 is not divisible by 7.
  • If we add the first and the 3rd digits of 418, we get twelve. Then if we subtract the second digit, one, we get eleven. Since we got a multiple of eleven, that means that our original number, 418, can be evenly divided by eleven.

Here is the factoring information for 418:

  • 418 is a composite number.
  • Prime factorization: 418 = 2 x 11 x 19
  • 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 418 has exactly 8 factors.
  • Factors of 418: 1, 2, 11, 19, 22, 38, 209, 418
  • Factor pairs: 418 = 1 x 418, 2 x 209, 11 x 38, or 19 x 22
  • 418 has no square factors that allow its square root to be simplified. √418 ≈ 20.445