Author: ivasallay

423 and Level 4

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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

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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

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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

 

417 and Level 6

/ ivasallay / 4 Comments

4 + 1 + 7 = 12, a multiple of three, so 417 can be evenly divided by three. Its other factors are listed below the puzzle.

The colors for today’s puzzle use the same key that puzzle 414 used and 422 will use, even though none of the squares here are blue. If I had colored the actual number 417, it would be blue. Also if I colored the number 414 it would have been green.

417 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-02

  • 417 is a composite number.
  • Prime factorization: 417 = 3 x 139
  • 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 417 has exactly 4 factors.
  • Factors of 417: 1, 3, 139, 417
  • Factor pairs: 417 = 1 x 417 or 3 x 139
  • 417 has no square factors that allow its square root to be simplified. √417 ≈ 20.4206

417 Logic

416 and Level 5

/ ivasallay / 2 Comments

416 is part of many Pythagorean triples. The smallest one is [160-384-416]. The largest one is the only primitive of the lot: [416-43,263-43,265]. Even though I find these triples incredibly easy to find, you can rest assured that you will probably go your entire life without ever having to know either of them!

416 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-02

  • 416 is a composite number.
  • Prime factorization: 416 = 2 x 2 x 2 x 2 x 2 x 13, which can be written 416 = (2^5) x 13
  • The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 x 2 = 12. Therefore 416 has exactly 12 factors.
  • Factors of 416: 1, 2, 4, 8, 13, 16, 26, 32, 52, 104, 208, 416
  • Factor pairs: 416 = 1 x 416, 2 x 208, 4 x 104, 8 x 52, 13 x 32, or 16 x 26
  • Taking the factor pair with the largest square number factor, we get √416 = (√16)(√26) = 4√26 ≈ 20.3961

 416 Logic

415 and Level 4

/ ivasallay / 2 Comments

415 is obviously a multiple of 5 so it is a composite number. Scroll down past the puzzle to view its other factors.

415 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-03-02

  • 415 is a composite number.
  • Prime factorization: 415 = 5 x 83
  • 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 415 has exactly 4 factors.
  • Factors of 415: 1, 5, 83, 415
  • Factor pairs: 415 = 1 x 415 or 5 x 83
  • 415 has no square factors that allow its square root to be simplified. √415 ≈ 20.3715

415 Logic

414 and Level 3

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414 is even and 4 + 1 + 4 = 9. These two facts help us to know that 414 can be evenly divided by 2, 3, 6, and 9. Its other factors are listed below the puzzle.

Today’s puzzle is really three puzzles in one. Solve some or all of them – your choice.

  • What are the missing factors?
  • Can you fill in the rest of the multiplication table?
  • What is the significance of the colors in the puzzle? Perhaps someone will type that answer in the comments because, otherwise, I’m not telling! (But you can look at puzzles 417 or 422 for more hints.)

414 Puzzle

Print the puzzles or type the factors on this black and white excel file: 10 Factors 2015-03-02

  • 414 is a composite number.
  • Prime factorization: 414 = 2 x 3 x 3 x 23, which can be written 414 = 2 x (3^2) x 23
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 x 3 x 2 = 12. Therefore 414 has exactly 12 factors.
  • Factors of 414: 1, 2, 3, 6, 9, 18, 23, 46, 69, 138, 207, 414
  • Factor pairs: 414 = 1 x 414, 2 x 207, 3 x 138, 6 x 69, 9 x 46, or 18 x 23
  • Taking the factor pair with the largest square number factor, we get √414 = (√9)(√46) = 3√46 ≈ 20.3470

 414 Factors