Math

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

408 and Level 6

/ ivasallay / Leave a comment

408 is divisible by 8 and so are 1408, 2408, 3408, 4408, . . . . . . and any other number whose last 3 digits are 408. This is true because if the last 3 digits of a number are divisible by 8, the entire number is also!

Also since 4 + 8 = 12, a multiple of 3, we know that 408 is divisible by 3.

408’s factors are listed below the puzzle.

408 Puzzle

Print the puzzles or type the factors on this excel file: 12 Factors 2015-02-23

  • 408 is a composite number.
  • Prime factorization: 408 = 2 x 2 x 2 x 3 x 17, which can be written 408 = (2^3) x 3 x 17
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 408 has exactly 16 factors.
  • Factors of 408: 1, 2, 3, 4, 6, 8, 12, 17, 24, 34, 51, 68, 102, 136, 204, 408
  • Factor pairs: 408 = 1 x 408, 2 x 204, 3 x 136, 4 x 102, 6 x 68, 8 x 51, 12 x 34, or 17 x 24
  • Taking the factor pair with the largest square number factor, we get √408 = (√4)(√102) = 2√102 ≈ 20.199

408 Logic

407 What’s the Logical Thing to Do, Spock?

/ ivasallay / 1 Comment

I was sad to learn of the death of Leonard Nimoy today. The factors of 407 will have to wait until the end of the post.

We only had one television when I was growing up, and my brothers insisted that it be tuned into Star Trek each week. Soon enough I became as interested as they were. After Star Trek was canceled, I loved watching Leonard Nimoy on Mission Impossible. I also loved every Star Trek movie he was ever in especially #2, 3, and 4. I was delighted that he was in the J. J. Abrams versions, and it’s painful to realize he will not be in any more.

When faced with any challenge, his character, Spock, always looked for the logical thing to do. I don’t imagine he would have wasted much time doing logic puzzles such as mine. Nevertheless, I dedicate this one to him.

THE FINAL FRONTIER

Print the puzzles or type the factors on this excel file: 12 Factors 2015-02-23

My husband and I didn’t watch Fringe until its second season when we heard that Leonard Nimoy would be in it. Then we watched it every week, often learning more about his character even though he only occasionally made an appearance.

We recently loved watching him in an old episode of Columbo even though he didn’t get away with his perfect crime.

Leonard Nimoy has died at the age of 83, a prime number for a man who always seemed to be in his prime.

On numerous occasions he has touched me, and his mind actually melded a little bit into mine. Some of who he was and what he did in life will permanently be a part of me.

Now here is 407’s factoring information:

  • 407 is a composite number.
  • Prime factorization: 407 = 11 x 37
  • 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 407 has exactly 4 factors.
  • Factors of 407: 1, 11, 37, 407
  • Factor pairs: 407 = 1 x 407 or 11 x 37
  • 407 has no square factors that allow its square root to be simplified. √407 ≈ 20.1742

407 Logic