Month: October 2015

654 and Level 5

/ ivasallay / Leave a comment

654 is the sum of the fourteen prime numbers from 19 to 73.

654 is also the hypotenuse of the Pythagorean triple 360-546-654. Which of 654’s factors is the greatest common factor of those three numbers?

654 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

—————————————————————————————————

  • 654 is a composite number.
  • Prime factorization: 654 = 2 x 3 x 109
  • 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 654 has exactly 8 factors.
  • Factors of 654: 1, 2, 3, 6, 109, 218, 327, 654
  • Factor pairs: 654 = 1 x 654, 2 x 327, 3 x 218, or 6 x 109
  • 654 has no square factors that allow its square root to be simplified. √654 ≈ 25.57342.

—————————————————————————————————

654 Logic

653 and Level 4

/ ivasallay / Leave a comment

When the last two digits of 653 are divided by 4, we get a remainder of 1 so there is an alternate way to tell if 653 is a prime number by first determining if it is the sum of any two perfect squares:

653 – 1 – 3 – 5 – 7 – 9 – 11 – 13 – 15 – 17 – 19 – 21 – 23 – 25 = 484 which is 22².

653 – 22² = 169 which is 13².

Thus 22² + 13² = 653, and 653 is the sum of two perfect squares.

From that mathematical statement we observe the following:

  • 22 and 13 have no common prime factors, and 653 is odd.
  • 653 is the hypotenuse of the primitive Pythagorean triple 315-572-653 which was calculated using 22² – 13², 2(22)(13), 22² + 13².
  • The only primitive Pythagorean triple hypotenuses less than √653 are 5, 13, and 17.
  • 653 obviously is not divisible by 5 or 13.
  • If 653 is not divisible by 17, it is a prime number. 653 is not divisible by 17.

 653 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

——————————————————————

  • 653 is a prime number.
  • Prime factorization: 653 is prime.
  • The exponent of prime number 653 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 653 has exactly 2 factors.
  • Factors of 653: 1, 653
  • Factor pairs: 653 = 1 x 653
  • 653 has no square factors that allow its square root to be simplified. √653 ≈ 25.55386.

How do we know that 653 is a prime number? If 653 were not a prime number, then it would be divisible by at least one prime number less than or equal to √653 ≈ 25.6. Since 653 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 653 is a prime number.

We also know that 653 is a prime number from the observations made at the beginning of this post.

—————————————————————————————————

653 Logic

652 and Level 3

/ ivasallay / Leave a comment

652 has 6 factors. 6 is a perfect number because all of its smaller factors, 1, 2, and 3, add up to its largest factor, 6.

The factors of 652 are 1, 2, 4, 163, and 326. The sum of those factors is 496, another perfect number. Note that all of 496’s smaller factors, 1, 2, 4, 8, 16, 31, 62, 124, and 248, add up to 496, its largest factor.

OEIS.org states that 652 is the only known non-perfect number that produces a perfect number in both of those situations.

652 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

—————————————————————————————————

  • 652 is a composite number.
  • Prime factorization: 652 = 2 x 2 x 163, which can be written 652 = (2^2) x 163
  • 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 652 has exactly 6 factors.
  • Factors of 652: 1, 2, 4, 163, 326, 652
  • Factor pairs: 652 = 1 x 652, 2 x 326, or 4 x 163
  • Taking the factor pair with the largest square number factor, we get √652 = (√4)(√163) = 2√163 ≈ 25.53429.

—————————————————————————————————

A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 12.)  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.

652 Factors

651 and Level 2

/ ivasallay / Leave a comment

21 x 31 = 651. Both of those factors are 5 away from their average, 26, so 651 is 25 less than 26² or 676.

The numbers in each of 651’s four factor pairs are odd, and the average of each and the distance each is from that average are both whole numbers. That means that 651 can be expressed as the difference of two squares four different ways. In this particular case the averages and distances generate four primitive Pythagorean triples with 651 as one of the legs:

  • 26² – 5² = 651; primitive triple 260-651-701
  • 50² – 43² = 651; primitive triple 651-4300-4349
  • 110² – 107² = 651; primitive triple 651-23540-23549
  • 326² – 325² = 651; primitive triple 651-211900-211901

651 is also a  pentagonal number.

651 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

—————————————————————————————————

  • 651 is a composite number.
  • Prime factorization: 651 = 3 x 7 x 31
  • 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 651 has exactly 8 factors.
  • Factors of 651: 1, 3, 7, 21, 31, 93, 217, 651
  • Factor pairs: 651 = 1 x 651, 3 x 217, 7 x 93, or 21 x 31
  • 651 has no square factors that allow its square root to be simplified. √651 ≈ 25.5147.

—————————————————————————————————

651 Factors

650 is the sum of all the clues in this Level 1 puzzle

/ ivasallay / 2 Comments

1² + 2² + 3² + 4² + 5² + 6² +7² + 8² + 9² +10² + 11² + 12²  = 650

Thus 650 is the 12th square pyramidal number and can be calculated using 12(12 +1)(2⋅12 + 1)/6.

If you add up all the clues in today’s Find the Factors puzzle, you will get the number 650. However, if you print the puzzle from the excel file, one of the clues is missing because it isn’t needed to find the solution.

650 is the hypotenuse of seven Pythagorean triples!

  • 72-646-650
  • 160-630-650
  • 182-624-650
  • 250-600-650
  • 330-560-650
  • 390-520-650
  • 408-506-650

Can you find the greatest common factor of each triple? Each greatest common factor will be one of the factors of 650 listed below the puzzle.

650 is the hypotenuse of so many Pythagorean triples because it is divisible by 5, 13, 25, 65, and 325 which are also hypotenuses of triples. The smallest three numbers to be the hypotenuses of at least 7 triples are 325, 425, and 650.

Since 25 x 26 = 650, we know that (25-1)(26 + 1) = 650 – 2. Thus 24 x 27 = 648.

650 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-19

—————————————————————————————————

  • 650 is a composite number.
  • Prime factorization: 650 = 2 x 5 x 5 x 13, which can be written 650 = 2 x (5^2) x 13
  • 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 650 has exactly 12 factors.
  • Factors of 650: 1, 2, 5, 10, 13, 25, 26, 50, 65, 130, 325, 650
  • Factor pairs: 650 = 1 x 650, 2 x 325, 5 x 130, 10 x 65, 13 x 50, or 25 x 26
  • Taking the factor pair with the largest square number factor, we get √650 = (√25)(√26) = 5√26 ≈ 25.495098.

650 Trees

—————————————————————————————————

649 and Level 6

/ ivasallay / Leave a comment

6 – 4 + 9 = 11 so 649 is divisible by 11.

649 is the short leg in exactly three Pythagorean triples. Can you determine which one is a primitive triple, and what are the greatest common factors of each of the two non-primitive triples?

  • 649-3540-3599
  • 649-19140-19151
  • 649-210600-210601

649 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-10-12

—————————————————————————————————

  • 649 is a composite number.
  • Prime factorization: 649 = 11 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 649 has exactly 4 factors.
  • Factors of 649: 1, 11, 59, 649
  • Factor pairs: 649 = 1 x 649 or 11 x 59
  • 649 has no square factors that allow its square root to be simplified. √649 ≈ 25.475478.

—————————————————————————————————

649 Logic

648 and Level 5

/ ivasallay / Leave a comment

648 is the sum of consecutive prime numbers 317 and 331.

The sixth root of 648 begins with 2.941682753. Notice all the digits from 1 to 9 appear somewhere in those nine decimal places. OEIS.org states that 648 is the smallest number that can make that claim.

From Archimedes-lab.org I learned some powerful facts about the number 648:

  • 16² – 17² + 18² – 19² + 20² – 21² +22² – 23² + 24² – 25² + 26² – 27² + 28² – 29² + 30² – 31² + 32² = 648
  • 48² – 47² + 46² – 45² + 44² – 43² +42² – 41² + 40² – 39² + 38² – 37² + 36² – 35² + 34² – 33² = 648
  • (1^2)(2^3)(3^4) = 648
  • 18² + 18²  = 648
  • (6^3) + (6^3) + (6^3) =648

648 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-10-12

—————————————————————————————————

  • 648 is a composite number.
  • Prime factorization: 648 = 2 x 2 x 2 x 3 x 3 x 3 x 3, which can be written 648 = (2^3) x (3^4)
  • The exponents in the prime factorization are 3 and 4. Adding one to each and multiplying we get (3 + 1)(4 + 1) = 4 x 5 = 20. Therefore 648 has exactly 20 factors.
  • Factors of 648: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 81, 108, 162, 216, 324, 648
  • Factor pairs: 648 = 1 x 648, 2 x 324, 3 x 216, 4 x 162, 6 x 108, 8 x 81, 9 x 72, 12 x 54, 18 x 36, or 24 x 27
  • Taking the factor pair with the largest square number factor, we get √648 = (√324)(√2) = 18√2 ≈ 25.455844122…

—————————————————————————————————

648 Logic

647 and Level 4

/ ivasallay / Leave a comment

647 is the sum of the five prime numbers from 113 to 139.

647 appears in only one Pythagorean triple, the primitive 647-209304-209305.

647 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-10-12

—————————————————————————————————

  • 647 is a prime number.
  • Prime factorization: 647 is prime.
  • The exponent of prime number 647 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 647 has exactly 2 factors.
  • Factors of 647: 1, 647
  • Factor pairs: 647 = 1 x 647
  • 647 has no square factors that allow its square root to be simplified. √647 ≈ 25.43619.

How do we know that 647 is a prime number? If 647 were not a prime number, then it would be divisible by at least one prime number less than or equal to √647 ≈ 25.4. Since 647 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 647 is a prime number.

—————————————————————————————————

647 Logic

646 and Level 3

/ ivasallay / Leave a comment

646 is the hypotenuse of the Pythagorean triple 304-570-646. What 2-digit number is the greatest common factor of those three numbers?

646 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-10-12

—————————————————————————————————

  • 646 is a composite number.
  • Prime factorization: 646 = 2 x 17 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 646 has exactly 8 factors.
  • Factors of 646: 1, 2, 17, 19, 34, 38, 323, 646
  • Factor pairs: 646 = 1 x 646, 2 x 323, 17 x 38, or 19 x 34
  • 646 has no square factors that allow its square root to be simplified. √646 ≈ 25.41653.

—————————————————————————————————

A Logical Approach to solve a FIND THE FACTORS puzzle: Find the column or row with two clues and find their common factor. (None of the factors are greater than 10.)  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.

646 Factors

645 Four Consecutive Odd Numbers Pass a Prime Number Test

/ ivasallay / Leave a comment

There is a quick test to see if an odd number is prime: Plug in the number for x in the equation y = 2^x (mod x). If y = 2, then x is VERY LIKELY a prime number. I call that equation the quick prime number test.

The smallest composite number that gives a false positive to this test is 341. Click on that number to see an in depth description of this quick prime number test and how it relates to Pascal’s triangle.

mod 341 calculator

This is only a picture of a calculator.

The second smallest number to give a false positive is 561. Click on that number to see MANY false positives for other tests for that composite number.

645 is not usually tested to see if it is prime because it ends with a 5 and the sum of its digits is a multiple of 3. From those two divisibility rules, we know that 645 is a composite number divisible by both 3 and 5.

Still 645 is the third smallest number to give a false positive to the quick prime number test.

Here is a chart of the quick prime number test applied to the odd numbers from 343 to 681. (A similar chart for odd numbers up to 341 can be seen here.) Whenever y equals 2, I’ve made that 2 red. Prime numbers have been highlighted in yellow. Pseudoprimes 561 and 645 are in bold print.

Prime Number Test 343-681

There are 119 numbers less than or equal to 561 that pass this prime number test, and only 3 of those numbers are composite numbers that give a false positive. Amazing.

Perhaps you noticed something else that’s pretty amazing: 641, 643, 645, and 647 are FOUR consecutive odd numbers that pass the quick prime number test, and 645 is the only composite number of the four. Those are the smallest four consecutive odd numbers that pass the test.

It makes me wonder if longer strings of numbers that pass the test are possible. As almost everybody knows 3, 5, and 7 are the only consecutive odd numbers that are also prime numbers. All other strings of three or more consecutive odd numbers contain at least one composite number that is a multiple of 3. Including pseudoprimes like 645 certainly opens up the possibility of longer strings of prime and pseudoprime numbers.

I also continue to be fascinated by the amount of times on the chart that y equals an odd power of 2 (2, 8, 32, 128, 512).

Interesting observation: All three of these pseudoprimes are of the form 4n + 1. All prime numbers of the form 4n + 1 can be written as the sum of two square numbers that have no common factors. 341, 561, and 645 cannot be written as such a sum so they cannot be prime numbers.

______________________________________

645 is the hypotenuse of the Pythagorean triple 387-516-645. What 3-digit number is the greatest common factor of those three numbers?

  • 645 is a composite number.
  • Prime factorization: 645 = 3 x 5 x 43
  • 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 645 has exactly 8 factors.
  • Factors of 645: 1, 3, 5, 15, 43, 129, 215, 645
  • Factor pairs: 645 = 1 x 645, 3 x 215, 5 x 129, or 15 x 43
  • 645 has no square factors that allow its square root to be simplified. √645 ≈ 25.39685.