Author: ivasallay

403 and Level 1

/ ivasallay / 1 Comment

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

/ ivasallay / 4 Comments

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

/ ivasallay / Leave a comment

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 Square Roots That Can Be Simplified

/ ivasallay / Leave a comment
  • 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.

399 and Level 5

/ ivasallay / Leave a comment

399 is 400 – 1. Since 20 x 20 = 400, it is VERY easy to know one of the factor pairs for 399. It will be 399 = (20 – 1) x (20 + 1).

399 Puzzle

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

  • 399 is a composite number.
  • Prime factorization: 399 = 3 x 7 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 399 has exactly 8 factors.
  • Factors of 399: 1, 3, 7, 19, 21, 57, 133, 399
  • Factor pairs: 399 = 1 x 399, 3 x 133, 7 x 57, or 19 x 21
  • 399 has no square factors that allow its square root to be simplified. √399 ≈ 19.975

399 Logic

398 and Level 4

/ ivasallay / Leave a comment

Let’s apply a couple of divisibility rules to the number 398 to find some of its factors. 2 is a factor of 398 because 8 is even. 4 is NOT a factor of 398 because 4 is NOT a factor of 98. We know this because 8 is divisible by 4, and 9 is an odd number.

398 Puzzle

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

  • 398 is a composite number.
  • Prime factorization: 398 = 2 x 199
  • 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 398 has exactly 4 factors.
  • Factors of 398: 1, 2, 199, 398
  • Factor pairs: 398 = 1 x 398 or 2 x 199
  • 398 has no square factors that allow its square root to be simplified. √398 ≈ 19.9499

398 Logic

397 and Level 3

/ ivasallay / Leave a comment

Let’s do a quick test to see if 397 is a prime number or a composite number: 2^397 (mod 397) = 2, so 397 is VERY LIKELY a prime number. Scroll down past the puzzle to know for sure.

397 Puzzle

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

  • 397 is a prime number.
  • Prime factorization: 397 is prime.
  • The exponent of prime number 397 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 397 has exactly 2 factors.
  • Factors of 397: 1, 397
  • Factor pairs: 397 = 1 x 397
  • 397 has no square factors that allow its square root to be simplified. √397 ≈ 19.9248588

How do we know that 397 is a prime number? If 397 were not a prime number, then it would be divisible by at least one prime number less than or equal to √397 ≈ 19.925. Since 397 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 397 is a prime number.

19² + 6² = 397 and 397 is the hypotenuse of primitive Pythagorean triple 228-325-397.

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.

397 Factors

396 What Pythagorean Triple Comes Next?

/ ivasallay / Leave a comment

396 is a multiple of 4, but not of 8, so just like 12, 20, 28, and 36, it is a leg of a primitive Pythagorean triple that is included in this infinite sequence of primitive triples (12–5–13), (20–21–29), (28–45–53), (36–77–85) . . . , which I’ve illustrated below:

20-21-29 What Pythagorean Triple Comes Next

Because they are Pythagorean triples, we know that 12² + 5² = 13², 20² + 21² = 29², 28² + 45² = 53², 36² + 77² = 85², and so forth.

What Pythagorean Triple Comes Next? Try to figure it out yourself, then scroll down a little bit to see if you are correct. In the meantime, let me tell you a little bit about the number 396:

  • 396 is a composite number.
  • Prime factorization: 396 = 2 x 2 x 3 x 3 x 11, which can be written 396 = (2^2) x (3^2) x 11
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 x 3 x 2 = 18. Therefore 396 has exactly 18 factors.
  • Factors of 396: 1, 2, 3, 4, 6, 9, 11, 12, 18, 22, 33, 36, 44, 66, 99, 132, 198, 396
  • Factor pairs: 396 = 1 x 396, 2 x 198, 3 x 132, 4 x 99, 6 x 66, 9 x 44, 11 x 36, 12 x 33 or 18 x 22
  • Taking the factor pair with the largest square number factor, we get √396 = (√11)(√36) = 6√11 ≈ 19.8997

The next primitive Pythagorean triple in the sequence can be illustrated like this:

11-117-125

Let me tell you about five Pythagorean triples in which 396 is one of the legs:

  • The answer to which Pythagorean triple comes next was (44–117–125), and is illustrated above. If we multiply that triple by 9, we get (396-1053-1125).
  • Because 396 equals 36 x 11, another triple can be found by multiplying the previous primitive in the sequence (36–77–85) by 11 to get (396-847-935).
  • If we multiply the first triple in the sequence (12–5–13) by 33, we get (396-165-429).
  • The 16th primitive triple in the sequence is (132–1085–1093). If we multiply it by 3 we get (396-3255-3279).
  • The 49th Pythagorean triple in our sequence of primitive triples above has a short leg that could be illustrated with 396 yellow squares. That primitive Pythagorean triple is (396–9797–9805).

395 and Level 2

/ ivasallay / Leave a comment

395 ends with a five so it is a composite number that is divisible by 5.

395 Puzzle

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

  • 395 is a composite number.
  • Prime factorization: 395 = 5 x 79
  • 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 395 has exactly 4 factors.
  • Factors of 395: 1, 5, 79, 395
  • Factor pairs: 395 = 1 x 395 or 5 x 79
  • 395 has no square factors that allow its square root to be simplified. √395 ≈ 19.875

395 Factors

Pythagorean Triples with 394

/ ivasallay / Leave a comment

Since 394 is an even number not divisible by 4, it can’t be part of a primitive Pythagorean triple. Scroll down past the puzzle to see its factors and its two non-primitive triples.

394 Puzzle

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

  • 394 is a composite number.
  • Prime factorization: 394 = 2 x 197
  • 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 394 has exactly 4 factors.
  • Factors of 394: 1, 2, 197, 394
  • Factor pairs: 394 = 1 x 394 or 2 x 197
  • 394 has no square factors that allow its square root to be simplified. √394 ≈ 19.849

To find Pythagorean triples with the number 394, first divide it in half to get 197. Since 197 is 14² + 1, we know it is the hypotenuse of the triple formed from 2(14 x 1), 14² – 1, and 14² + 1 or (28-195-197). If we multiply that triple by 2, we get (56-390-394).

Prime number 197 is also the short leg of the primitive Pythagorean triple, (197-19404-19405). Doubling each element of that triple gives us (394-38808-38810).

394 Factors