factor

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

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

/ ivasallay / Leave a comment

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

413 and Level 2

/ ivasallay / 2 Comments

413 can’t be divided evenly by 2, 3, or 5. Can it be divided by 7? Here’s a couple of quick divisibility tricks to try, each requiring the last number to be separated from the rest giving us 41 and 3.

  1. Double the 3 and subtract it from 41. That gives us 41 – 6 = 35, a multiple of 7 so 413 can be divided evenly by 7.
  2. Multiply the 3 by five and add it to 41. That gives us 41 + 15 = 56, a multiple of 7 as well, so we can draw the same conclusion.

Thank you to abyssbrain for telling me about the second divisibility trick. All of 413’s factors are listed below the puzzle.

413 Puzzle

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

  • 413 is a composite number.
  • Prime factorization: 413 = 7 x 59
  • 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 413 has exactly 4 factors.
  • Factors of 413: 1, 7, 59, 413
  • Factor pairs: 413 = 1 x 413 or 7 x 59
  • 413 has no square factors that allow its square root to be simplified. √413 ≈ 20.3224

413 Factors

412 and Level 1

/ ivasallay / Leave a comment

Because 12 is divisible by 4, we know that 412 is also divisible by 4.

The number 12 has six factors. Does 412 have more than that? Find out after the puzzle!

412 Puzzle

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

  • 412 is a composite number.
  • Prime factorization: 412 = 2 x 2 x 103, which can be written 412 = (2^2) x 103
  • 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 412 has exactly 6 factors.
  • Factors of 412: 1, 2, 4, 103, 206, 412
  • Factor pairs: 412 = 1 x 412, 2 x 206, or 4 x 103
  • Taking the factor pair with the largest square number factor, we get √412 = (√4)(√103) = 2√103 ≈ 20.2978

Surprised?

412 Factors

409 Happy Birthday, John!

/ ivasallay / 4 Comments

409 = 20² + 3² so it is the hypotenuse of a Pythagorean triple. That triple is 120-391-409. Could 409 possibly be a prime number? The answer is at the end of the post.

Today is my son’s birthday. He lives on the other side of the country, so I’m making him a puzzle cake that he can devour. Many years ago when he still lived at home, he could solve any of my puzzles, so I know he can handle this one even though I haven’t revealed its difficulty level.

409 Puzzle

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

I am grateful for facebook! His wife posted, “I felt like I was wrapping presents for a 10 year old boy this morning.” My son is a young father who loves, loves, loves Legos.

  • 409 is a prime number.
  • Prime factorization: 409 is prime.
  • The exponent of prime number 409 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 409 has exactly 2 factors.
  • Factors of 409: 1, 409
  • Factor pairs: 409 = 1 x 409
  • 409 has no square factors that allow its square root to be simplified. √409 ≈ 20.2237

How do we know that 409 is a prime number? If 409 were not a prime number, then it would be divisible by at least one prime number less than or equal to √409 ≈ 20.2237. Since 409 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 409 is a prime number.

409 Logic