Prime factorization

40% of Numbers Up To 400 Have Square Roots That Can Be Simplified

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  • 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

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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

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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

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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?

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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

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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

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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

393 and A Chance to Solve a Problem

/ ivasallay / 2 Comments

A problem is a chance for you to do your best

A few weeks ago Brilliant Quotes tweeted a quote from Duke Ellington: “A problem is a chance for you to do your best.”

I immediately thought of problems students are given in mathematics classes because this quote applies to those problems.

It also applies to any other kind of problem a person may face. Sometimes those problems are much more challenging than problems involving numbers. All problems are opportunities for us to do our best.

Let’s find the factoring information for a relatively easy number – 393.

  • 393 is a composite number.
  • Prime factorization: 393 = 3 x 131
  • 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 393 has exactly 4 factors.
  • Factors of 393: 1, 3, 131, 393
  • Factor pairs: 393 = 1 x 393 or 3 x 131
  • 393 has no square factors that allow its square root to be simplified. √393 ≈ 19.824

Is 393 in any Pythagorean triples?

  • 393-524-655  which is [3 – 4 – 5] times 131
  • 393-25740-25743 which is [131-8580-8581] times 3
  • Primitive 393-8576-8585
  • and Primitive 393-77224-77225

At the end of his post a-promise-broken, Established1962 tells a personal and very funny story involving his copy of Duke Ellington’s autobiography, “Music Is My Mistress.” The quote above is included in that book.

 

 

392 Happy Valentine’s Day

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392 is divisible by seven because 39 minus two times the last digit is 35, a multiple of seven. That is one of the reasons 392 is a composite number. All of its factors are listed below the puzzle.

392 Puzzle

Print the puzzles or type the factors on this excel file:12 Factors 2015-02-09

  • 392 is a composite number.
  • Prime factorization: 392 = 2 x 2 x 2 x 7 x 7, which can be written 392 = (2^3) x (7^2)
  • The exponents in the prime factorization are 3 and 2. Adding one to each and multiplying we get (3 + 1)(2 + 1) = 4 x 3 = 12. Therefore 392 has exactly 12 factors.
  • Factors of 392: 1, 2, 4, 7, 8, 14, 28, 49, 56, 98, 196, 392
  • Factor pairs: 392 = 1 x 392, 2 x 196, 4 x 98, 7 x 56, 8 x 49, or 14 x 28
  • Taking the factor pair with the largest square number factor, we get √392 = (√2)(√196) = 14√2 ≈ 19.799

392 Factors

391 To a Wild Rose

/ ivasallay / 2 Comments

2^391 (mod 391) = 179, not 2 so 391 is definitely a composite number. Scroll down below the puzzle to view its factors.

When I was in elementary school, my sister, Sue, taught me a song she learned in her junior high choir class. I’m not able to find these lyrics on the internet, but as best as I can remember, the words of the song were:

In the woods, in the spring, blooms the lovely wild rose. Every bud waking facing the sun. Hearts of spring, beat on thee, as the light of day grows. Dew drops glisten on you, Wild Rose. Fill the countryside with your lovely fragrance. Sun keeps your petals warm, leaves hide you from harm. Day will break, petals close, close them, oh, so tightly. Cover your beauty from the cool night. Birds, winging, singing through the woodland. Sleep and dream, sleep and dream, sleep and dream.

391 Puzzle

Print the puzzles or type the factors on this excel file:12 Factors 2015-02-09

  • 391 is a composite number.
  • Prime factorization: 391 = 17 x 23
  • 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 391 has exactly 4 factors.
  • Factors of 391: 1, 17, 23, 391
  • Factor pairs: 391 = 1 x 391 or 17 x 23
  • 391 has no square factors that allow its square root to be simplified. √391 ≈ 19.774

  • Note that 17 + 3 = 20, and 20 + 3 = 23, and (20^2) – (3^2) = 400 – 9 = 391.
  • 2(3*20), (20^2) – (3^2), (20^2) + (3^2) makes primitive Pythagorean triple 120-391-409.
  • 391-76440-76441 is another primitive Pythagorean triple
  • 184-345-391 is [8-15-17] multiplied by 23
  • 391-3312-3335 is [17-144-145] multiplied by 23
  • 391-4488-4505 is [23-264-265] multiplied by 17

391 is in this cool pattern:

391 Factors