Author: ivasallay

415 and Level 4

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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

413 and Level 2

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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

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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

411 Some Things You Should Know Before You Start a Blog

/ ivasallay / 26 Comments

You’re passionate about some subject and are considering starting a blog. Here’s some things you should know first:

What your blog is about will eventually change to one degree or another. I have noticed many bloggers redefine themselves, some more than others. My blog was initially a way to share a factoring puzzle I invented, but most of the people who visit my blog are not looking for puzzles; they want to find the factors of a particular number. After all, my blog name implies that factors can be found here. To satisfy these viewers, I now give the factoring information for the number 411:

  • 411 is a composite number.
  • Prime factorization: 411 = 3 x 137
  • 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 411 has exactly 4 factors.
  • Factors of 411: 1, 3, 137, 411
  • Factor pairs: 411 = 1 x 411 or 3 x 137
  • 411 has no square factors that allow its square root to be simplified. √411 ≈ 20.2731

When I first noticed that people were visiting my blog because they wanted to know the factors of a number, I resisted changing, but later I gave in. I even edited all my previous blog posts to include factored numbers. Eventually I added the square roots of numbers and indicated if those square roots could be simplified. Still I keep my original intent and publish the puzzles I create.

Coming up with imaginative titles for EVERY blog post can be tedious or even impossible. Many people judge whether or not a post is worth reading by the title. Sometimes a great title pops into my head, but most of the time I’ve got nothing. I started giving my puzzle posts boring titles like “Level 3”  and actually had several posts with that same name. When my stats page told me that someone liked or visited a post called Level 2, I wasn’t sure which post that was. When I started factoring numbers, it solved my problem of giving each post a distinctive title even if it was still a boring name. Now I often use unimaginative titles like “406 and Level 4”.

Blogging uses up more time than you imagine it will. I once read a very long, hilariously funny, grammatically perfect post that the author claimed he wrote during his lunch time. I find that claim very hard to believe because it certainly doesn’t work that way for me. I can spend 15-30 minutes reading or writing blogs, but the clock at my house indicates that I actually spent two hours or more. No matter what clock I use, it’s the same story.

If you write a blog, you will connect with a community of people who read your blog and you read theirs. All that reading takes time. Commenting on someone else’s blog or answering a comment on your own blog also takes time. It may seem like you’re only spending a few minutes at a time, but it really is much longer than that. Family members may complain about other things not getting done or quality time not being spent even if they are trying to be supportive. You will find blogs you want to read even though its writer doesn’t feel the same way about your blog. You will learn so much reading other people’s blogs, and you may want to read more books about a particular subject or even write about it yourself. You will find yourself wishing there were more hours in the day, or that you could actually make some money blogging and cut back on your hours at your regular job.

Most bloggers don’t make money blogging. If you want to make money on WordPress, you have to have enough views, you have to purchase your own domain name, and you have to sign up for WordAds. After some consideration, I took this step in August 2014. I registered my domain name for $18 a year and signed up for WordAds. I also pay another $8 each year to keep my contact information private.

How many views do you have to have to make money on WordPress? Update: Since writing this post, WordPress allows ALL of its bloggers to sign up for Wordads. Now every blog with a registered domain name is a potential money maker, even if it turns out to be just a little bit of money each month. The amount of money that you’ll make will depend on how many views you have and where those views come from. 

Was it worth it to register your own domain name? Yes, the number of views I’ve received after registering is much, much greater than it was before I registered. It is much easier to tell friends and acquaintances to look for Findthefactors.com than it was to tell them that name with wordpress added to it. Even though it cost more money than I might have made back, it helps me reach more viewers, and that helps me meet my original objective. The only drawback is that if don’t renew my registration every year Findthefactors.com could become what they call a parking lot. I would have no control over what people would see if they went to the site. If you read the fine print when you register your domain, this stipulation is made quite clear. Somebody else could put whatever they want on the unregistered site, including adult content. I intend to continue registering Findthefactor.com for the rest of my life. Hopefully my children and grandchildren will also after I’m gone.

If you’re still thinking of starting a blog, do it because you love a particular subject. Do it because it pleases you. There will be days when absolutely nobody else looks at your blog. You will gain some followers, but some of those followers will get discouraged with their statistics and quit blogging and quit reading your blog. You will also get some followers who follow you for the sole purpose of trying to get you to follow them back. Once you follow them, they never look at your blog again. If you blog, you have to do it for yourself, and if anyone else enjoys it, that is a bonus.

I chose WordPress because it allows me to put excel documents on my blog. (Blogspot didn’t.) Unfortunately people who view my blog on their smart phones can’t see those documents. I hope there are a few teachers out there who at least occasionally look at my blog on their home or school computers and print out the puzzles for their classes. That would make my decision to add those documents definitely worth it.

Fellow bloggers, what other things do you think new bloggers should know?

410 Is The Sum of Two Squares Two Different Ways

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410 = 59 + 61 + 67 + 71 + 73 + 79, six consecutive prime numbers.

410 = 199 + 211, two consecutive prime numbers.

  • 410 is a composite number.
  • Prime factorization: 410 = 2 x 5 x 41
  • 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 410 has exactly 8 factors.
  • Factors of 410: 1, 2, 5, 10, 41, 82, 205, 410
  • Factor pairs: 410 = 1 x 410, 2 x 205, 5 x 82, or 10 x 41
  • 410 has no square factors that allow its square root to be simplified. √410 ≈ 20.2485

Those factors hardly seem like anything to write home about, but wait….

410 equals

How do I know all this? I applied something I read on Dr. David Mitchell’s blog, Lattice Labyrinths. He recently wrote a fascinating post about Tessellations and Pythagorean Triples. You have to read the article or at least look at his tessellations!

I didn’t know multiplying tessellating lattice labyrinths had anything to do with Pythagorean triples, but they do! (Actually I didn’t know anything about tessellating lattice labyrinths until I read the post.)

About the fourth or fifth paragraph of Dr. Mitchell’s post, he states something I didn’t know before: If you take one number that is the sum of two squares and find another number that is also the sum of two squares, and then if you multiply those two numbers together, that product will also be the sums of two squares! Since I knew both 10 and 41 could be written as the sum of two squares, I had to see if 410 could be also. I was doubly surprised and certainly not the least bit disappointed. I don’t know if it’s unusual that most of 410’s factors can also be written as the sum of two squares.

Because 410 and so many of its factors have that property, it is the hypotenuse of four non-primitive Pythagorean triples:

  • [266-312-410] is [133 – 156 – 205] times 2.
  • [246-328-410] is [3-4-5] times 82.
  • [168-374-410] is [84-187-205] times 2.
  • [90-400-410] is [9-40-41] times 10.

It is also the short leg of four other triples:

  • [410-984-1066] is [5-12-13] times 82.
  • [410-1656-1706] is [205-828-853] times 2.
  • [410-8400-8410] is [41-840-841] times 10.
  • [410-42024-42026] is [205-21012-21013] times 2.

 

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

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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

406 and Level 4

/ ivasallay / 4 Comments

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