409 Happy Birthday, John!

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

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408 and Level 6

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?

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

406 and Level 4

Let’s apply the 7 divisibility trick to 406.

  • Separate the last digit from the rest and multiply that last digit by two: 6 x 2 = 12.
  • Now find the difference between the remaining number, 40 and 12: 40 – 12 = 28.
  • Since 28 is a multiple of 7, we know that 406 is also divisible by 7.

406 Puzzle

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

  • 406 is a composite number.
  • Prime factorization: 406 = 2 x 7 x 29
  • 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 406 has exactly 8 factors.
  • Factors of 406: 1, 2, 7, 14, 29, 58, 203, 406
  • Factor pairs: 406 = 1 x 406, 2 x 203, 7 x 58, or 14 x 29
  • 406 has no square factors that allow its square root to be simplified. √406 ≈ 20.1494

406 Logic

405 and Level 3

405 times 2 equals 810, and 810 divided by 10 is 81. And guess what, 405 divided by 5 is also 81. Multiplying by .2 instead of dividing by 5 makes a nice division trick!

All of 405’s factors are listed below the puzzle.

405 Puzzle

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

  • 405 is a composite number.
  • Prime factorization: 405 = 3 x 3 x 3 x 3 x 5, which can be written 405 = (3^4) x 5
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 x 2 = 10. Therefore 405 has exactly 10 factors.
  • Factors of 405: 1, 3, 5, 9, 15, 27, 45, 81, 135, 405
  • Factor pairs: 405 = 1 x 405, 3 x 135, 5 x 81, 9 x 45, or 15 x 27
  • Taking the factor pair with the largest square number factor, we get √405 = (√81)(√5) = 9√5 ≈ 20.1246

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.

405 Factors

404 and Level 2

It may seem like trivial trivia, but every factor of 404 is a palindrome. Scroll down past the puzzle to find out what all those factors are.

404 Puzzle

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

  • 404 is a composite number.
  • Prime factorization: 404 = 2 x 2 x 101, which can be written 404 = (2^2) x 101
  • 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 404 has exactly 6 factors.
  • Factors of 404: 1, 2, 4, 101, 202, 404
  • Factor pairs: 404 = 1 x 404, 2 x 202, or 4 x 101
  • Taking the factor pair with the largest square number factor, we get √404 = (√4)(√101) = 2√101 ≈ 20.0998

404 Factors

403 and Level 1

Madam, I’m Adam! There is something palindromic about the factors of 403. What could they possibly be? Scroll down past the puzzle to see what they are.

403 Puzzle

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

  • 403 is a composite number.
  • Prime factorization: 403 = 13 x 31
  • 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 403 has exactly 4 factors.
  • Factors of 403: 1, 13, 31, 403
  • Factor pairs: 403 = 1 x 403 or 13 x 31
  • 403 has no square factors that allow its square root to be simplified. √403 ≈ 20.0749

403 Factors

402 and Passing Along The Spectacular Blog Award

We can find some of the factors of 402 rather quickly. It’s even, so 2 is a factor. 4 + 0 + 2 = 6, a multiple of 3, so 3 is also a factor. Six, then, would also be a factor. How many factors does 402 have and what are they? Scroll down to the end of the post to find out!

Nerd in the Brain created the-Spectacular-Blog-Award. This is how she describes it: “🙂It’s a simple award with no strings attached…no questions to answer, no specific number of people to nominate, no obligation. Recipients can just bask in the glory of knowing that another blogger thinks they’re super-awesome.” 🙂 

She continued, “This award is not just for me to give to people! If you want to let another blogger know that you think they’re fantastic, you go right ahead and snag this little award and pass it along to them. Share, share!” Keep Nerd in the Brain in the loop if you give someone this award.

So I for one am “snagging” this award to pass on to someone else:

Update: I gave the award to Abyssbrain whose blog, Mathemagicalsite.wordpress.com, has, unfortunately, been deleted.

The only thing I can think of to quell my disappointment is to find a couple more things about the number 402 and add them to this post:

Here are five ways 402 is the sum of three squares:

  • 20² + 1² + 1² = 402
  • 19² + 5² + 4² = 402
  • 17² + 8² + 7² = 402
  • 16² + 11² + 5² = 402
  • 13² + 13² + 8² = 402

402 is 123 in BASE 19 because 1(19²) + 2(19¹) + 3(19°) = 402.

  • 402 is a composite number.
  • Prime factorization: 402 = 2 x 3 x 67
  • 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 402 has exactly 8 factors.
  • Factors of 402: 1, 2, 3, 6, 67, 134, 201, 402
  • Factor pairs: 402 = 1 x 402, 2 x 201, 3 x 134, or 6 x 67
  • 402 has no square factors that allow its square root to be simplified. √402 ≈ 20.0499

401 and Level 6

We can easily see that 401 is not divisible by 2, 3, or 5. Is it a prime number? Let’s do a quick test: 2^401 (mod 401) = 2, so 401 is VERY LIKELY a prime number. Scroll down past the puzzle to know for sure.

401 Puzzle

Print the puzzles or type the factors on this excel file: 10 Factors 2015-02-16

  • 401 is a prime number.
  • Prime factorization: 401 is prime.
  • The exponent of prime number 401 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 401 has exactly 2 factors.
  • Factors of 401: 1, 401
  • Factor pairs: 401 = 1 x 401
  • 401 has no square factors that allow its square root to be simplified. √401 ≈ 20.025

How do we know that 401 is a prime number? If 401 were not a prime number, then it would be divisible by at least one prime number less than or equal to √401 ≈ 20.025. Since 401 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 401 is a prime number.

Pythagorean triples: Since 401 is a prime number, it can only be a part of primitive Pythagorean triples: 40-399-401 and 401-80400-80401.

401 Logic

40% of Numbers Up To 400 Have Reducible Square Roots

  • 400 is a composite number.
  • Prime factorization: 400 = 2 x 2 x 2 x 2 x 5 x 5, which can be written 400 = (2^4) x (5^2)
  • The exponents in the prime factorization are 4 and 2. Adding one to each and multiplying we get (4 + 1)(2 + 1) = 5 x 3 = 15. Therefore 400 has exactly 15 factors.
  • Factors of 400: 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400
  • Factor pairs: 400 = 1 x 400, 2 x 200, 4 x 100, 5 x 80, 8 x 50, 10 x 40, 16 x 25, or 20 x 20
  • 400 is a perfect square. √400 = 20

A few months ago I made a chart showing the number of factors for the first 300 counting numbers. Since this is my 400th post, I’d like to include a chart showing the number of factors for all the numbers from 301 to 400. I’m also interested in consecutive numbers with the same number of factors and whether or not the square root of a number can be reduced. The red numbers have square roots that can be reduced.

301-400 Same Number of Factors

 

The longest streak of consecutive numbers with the same number of factors is only three. There are three sets of three consecutive numbers on this chart. (Between 200 and 300 there was a streak of four consecutive numbers with six factors each.)

How do the number of factors of these 100 numbers stack up against the previous 300? The following chart shows the number of integers with a specific number of factors and how many of those integers have reducible square roots:

1-400 Number of Factors

  • 39.5% or slightly less than 40% of the numbers up to 400 have reducible square roots.
  • Most of these numbers have 2, 4, or 8 factors. Numbers with two factors are prime numbers. Almost all numbers with four factors are the product of two different prime numbers, and nearly two-thirds of the numbers with eight factors are the product of three different prime numbers.
  • There isn’t much change between the percentages of reducible square roots from one list to the next.