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

/ ivasallay / Leave a comment

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

390 and Level 4

/ ivasallay / Leave a comment

390 is a multiple of 6 because it is even and all of its digits are divisible by 3. Scroll down past the puzzle to view its factors.

390 Puzzle

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

  • 390 is a composite number.
  • Prime factorization: 390 = 2 x 3 x 5 x 13
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 x 2 x 2 x 2 = 16. Therefore 390 has exactly 16 factors.
  • Factors of 390: 1, 2, 3, 5, 6, 10, 13, 15, 26, 30, 39, 65, 78, 130, 195, 390
  • Factor pairs: 390 = 1 x 390, 2 x 195, 3 x 130, 5 x 78, 6 x 65, 10 x 39, 13 x 30, or 15 x 26
  • 390 has no square factors that allow its square root to be simplified. √390 ≈ 19.748

390 Logic

389 How to Know if a Prime Number is the Hypotenuse in a Primitive Pythagorean Triple

/ ivasallay / 2 Comments
  • 389 is a prime number.
  • Prime factorization: 389 is prime.
  • The exponent of prime number 389 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 389 has exactly 2 factors.
  • Factors of 389: 1, 389
  • Factor pairs: 389 = 1 x 389
  • 389 has no square factors that allow its square root to be simplified. √389 ≈ 19.723

How do we know that 389 is a prime number? If 389 were not a prime number, then it would be divisible by at least one prime number less than or equal to √389 ≈ 19.723. Since 389 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, or 19, we know that 389 is a prime number.

If a Pythagorean triple is a primitive, then its hypotenuse is always an odd number. Sometimes it is also a prime number. How can we know if a particular prime number, such as 389, is ever the hypotenuse of a Pythagorean triple?

Divide the last two digits of the prime number by 4 and look at the remainder:

  • If the remainder is 3, then that prime number will not ever be the hypotenuse of any Pythagorean triple.
  • If the remainder is 1, then it is the hypotenuse of a primitive Pythagorean triple! Find the proof here.

The last two digits of 389 is 89, and 89 ÷ 4 = 22 R1, so 389 and every other prime number ending in 89 is the hypotenuse of a primitive Pythagorean triple. What are the other two sides of this particular triangle? First we will figure out what two square numbers equal 389, then we will use those two numbers to figure out what two square numbers equal 389 squared:

Have you ever noticed that 1 is the first square number, 1 + 3 or 4 is also a square number, and 1 + 3 + 5 or 9 is yet another square number? This principle can be summarized by writing 1 + 3 + 5 + . . . + (2n – 1) = n squared.

Square Numbers - Sums of First Odd Numbers`

I’ll use this fact to determine when 389 is the hypotenuse of a primitive Pythagorean triple. I’ll subtract 1 from 389, then 3 from that result, and continue to subtract the next odd number until I get a perfect square. I should find the perfect square before I reach 194 (about half of 389).

So let’s begin:

389 to 289

At this point I know that 17² + another square equals 389. There are a few methods I can use to find the other perfect square.

  • I can find the other square by subtracting: 389 – 17²
  • I can take the last odd number used, 19, add one, then divide by two and then square that number.
  • Or I can continue subtracting odd numbers as before:

289 to 100

All of these methods give us 100 or 10² as the second perfect square. Notice that (33 + 1)/2 = 17 as well.

Now we have enough information to determine the other two numbers that make up the Pythagorean triple with 389 as the hypotenuse.

  • All of the numbers, a – b – c, in the triple can be found using the formulas Euclid provided more than 2000 years ago. Let 10 = n; and 17 = m.
  • a = m² – n² = 17² – 10² = 189 
  • b = 2mn = 2(10)(17) = 340
  • c = m² + n² = 17² + 10² = 389
  • thus 189 – 340 – 389 make a primitive Pythagorean triple. We know it is a primitive because 389 is prime and has no other factors besides one and itself.

388 and Level 3

/ ivasallay / Leave a comment

88 is divisible by 4 so 388 is also divisible by 4 and is a composite number. 88 is also divisible by 8, but the third to the last digit is odd, making 388 NOT divisible by 8.

388 Puzzle

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

  • 388 is a composite number.
  • Prime factorization: 388 = 2 x 2 x 97, which can be written 388 = (2^2) x 97
  • 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 388 has exactly 6 factors.
  • Factors of 388: 1, 2, 4, 97, 194, 388
  • Factor pairs: 388 = 1 x 388, 2 x 194, or 4 x 97
  • Taking the factor pair with the largest square number factor, we get √388 = (√4)(√97) = 2√97 ≈ 19.698

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.

388 Factors

387 and Level 2

/ ivasallay / Leave a comment

Is 387 prime or composite? Well, 8 + 7 = 15, a multiple of 3, so 387 is a composite number divisible by 3.

Here’s sending a little luv your way!

 387 Puzzle

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

  • 387 is a composite number.
  • Prime factorization: 387 = 3 x 3 x 43, which can be written 387 = (3^2) x 43
  • 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 387 has exactly 6 factors.
  • Factors of 387: 1, 3, 9, 43, 129, 387
  • Factor pairs: 387 = 1 x 387, 3 x 129, or 9 x 43
  • Taking the factor pair with the largest square number factor, we get √387 = (√9)(√43) = 3√43 ≈ 19.672

387 Factors

386 and Level 1

/ ivasallay / 2 Comments

386 is even and composite.

386 Puzzle

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

  • 386 is a composite number.
  • Prime factorization: 386 = 2 x 193
  • 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 386 has exactly 4 factors.
  • Factors of 386: 1, 2, 193, 386
  • Factor pairs: 386 = 1 x 386 or 2 x 193
  • 386 has no square factors that allow its square root to be simplified. √386 ≈ 19.647

386 Factors

385 Is a Square Pyramidal Number

/ ivasallay / 3 Comments

385 is the sum of the squares of the first ten counting numbers. Let me demonstrate what that means. All of the following are square pyramidal numbers:

  • 1² = 1
  • 1² + 2² = 5
  • 1² + 2² + 3² = 14
  • 1² + 2² + 3² + 4² = 30
  • 1² + 2² + 3² + 4² + 5² = 55
  • 1² + 2² + 3² + 4² + 5² + 6² = 91
  • 1² + 2² + 3² + 4² + 5² + 6² + 7² = 140
  • 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² = 204
  • 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² = 285
  • 1² + 2² + 3² + 4² + 5² + 6² + 7² + 8² + 9² + 10² = 385

We could have found the total, 385, much more quickly by putting the number ten in for “n” in this formula

Which means the sum of the first ten square counting numbers = (10 x 11 x 21)/6 = 5 x 11 x 7 = 385. (Notice its prime factorization!)

Here’s a video explanation:

Thanks to solvemymaths for bringing the video to my attention. It clearly explains why the formula works.

Here is the factoring information for 385:

  • 385 is a composite number.
  • Prime factorization: 385 = 5 x 7 x 11
  • 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 385 has exactly 8 factors.
  • Factors of 385: 1, 5, 7, 11, 35, 55, 77, 385
  • Factor pairs: 385 = 1 x 385, 5 x 77, 7 x 55, or 11 x 35
  • 385 has no square factors that allow its square root to be simplified. √385 ≈ 19.621

384 and Level 6

/ ivasallay / Leave a comment

Today’s Puzzle:

384 Puzzle

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

Here is a logical way to use the clues in the puzzle to find the solution:

384 Logic

Factors of 384:

384 is even and, therefore, composite.

  • 384 is a composite number.
  • Prime factorization: 384 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 3, which can be written 384 = (2^7) x 3
  • The exponents in the prime factorization are 7 and 1. Adding one to each and multiplying we get (7 + 1)(1 + 1) = 8 x 2 = 16. Therefore 384 has exactly 16 factors.
  • Factors of 384: 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 128, 192, 384
  • Factor pairs: 384 = 1 x 384, 2 x 192, 3 x 128, 4 x 96, 6 x 64, 8 x 48, 12 x 32, or 16 x 24
  • Taking the factor pair with the largest square number factor, we get √384 = (√6)(√64) = 8√6 ≈ 18.596

Sum-Difference Puzzles:

6 has two factor pairs. One of those pairs adds up to 5, and the other one subtracts to 5. Put the factors in the appropriate boxes in the first puzzle.

384 has eight factor pairs. One of the factor pairs adds up to 40, and a different one subtracts to 40. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

More about the Number 384:

384 is in this cool pattern: