Level 4 Puzzle

713 and Level 4

/ ivasallay / Leave a comment
  • 713 is a composite number.
  • Prime factorization: 713 = 23 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 713 has exactly 4 factors.
  • Factors of 713: 1, 23, 31, 713
  • Factor pairs: 713 = 1 x 713 or 23 x 31
  • 713 has no square factors that allow its square root to be simplified. √713 ≈ 26.7020598.

Go ahead and give this Level 4 puzzle a try:

 

713 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-14

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Would you like to know a little more about 713?

713 can be written as the sum of consecutive numbers several ways:

  • 356 + 357 = 713; that’s two consecutive numbers.
  • 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 = 713; that’s 23 consecutive numbers.
  • 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 = 713; that’s 31 consecutive numbers.

713 is also the sum of three consecutive prime numbers: 233 + 239 + 241 = 713.

Two of its factors are 4 numbers away from their average, 27, so (23)(31) = (27 – 4)(27 + 4) = 27² – 4² = 713

713 is palindrome NN in BASE 30 (N = 23 in base 10); note that 23(30) +23(1) = 713.

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713 Logic

Thank you, Ricardo, for tweeting the puzzle and your work:

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707 and Level 4

/ ivasallay / Leave a comment
  • 707 is a composite number.
  • Prime factorization: 707 = 7 x 101
  • 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 707 has exactly 4 factors.
  • Factors of 707: 1, 7, 101, 707
  • Factor pairs: 707 = 1 x 707 or 7 x 101
  • 707 has no square factors that allow its square root to be simplified. √707 ≈ 26.58947.

Here is today’s factoring puzzle:

707 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-12-07

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Here are a few more facts about 707:

707 is a palindrome in base 10 and in base 100.

707 is the sum of seven consecutive numbers: 98 + 99 + 100 + 101 + 102 + 103 + 104 = 707

It is also the sum of two consecutive numbers: 353 + 354 = 707.

707 is the sum of these five consecutive prime numbers: 131 + 137 + 139 + 149 + 151 = 707.

And it is the sum of the nineteen prime numbers from 5 to 73.

Since 101 is one of its prime factors, 707 is the hypotenuse of Pythagorean triple 140-693-707. What is the greatest common factor of those three numbers?

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707 Logic

Level 4 Christmas Puzzle #699

/ ivasallay / Leave a comment
  • 699 is a composite number.
  • Prime factorization: 699 = 3 x 233
  • 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 699 has exactly 4 factors.
  • Factors of 699: 1, 3, 233, 699
  • Factor pairs: 699 = 1 x 699 or 3 x 233
  • 699 has no square factors that allow its square root to be simplified. √699 ≈ 26.438608.

Here is a Christmas puzzle for you to solve. It’s numbered 699 to distinguish it from every other puzzle I make:

699 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-30

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Here are a few thoughts I’ve had about the number 699:

699 is the smallest number whose digits add up to 24.

Every odd number greater than 1 is the sum of 2 consecutive numbers. 699 is the sum of 349 and 350.

Every number that is divisible by 3 is the sum of 3 consecutive numbers: 232 + 233 + 234 = 699.

Also 699 is the hypotenuse of Pythagorean triple 315-624-699. Which factor of 699 is the greatest common factor of those three numbers?

699 is palindrome 272 in BASE 17; note that 2(289) + 7(17) + 2(1) = 699

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699 Logic

692 Happy Thanksgiving!

/ ivasallay / 2 Comments
  • 692 is a composite number.
  • Prime factorization: 692 = 2 x 2 x 173, which can be written 692 = (2^2) x 173
  • 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 692 has exactly 6 factors.
  • Factors of 692: 1, 2, 4, 173, 346, 692
  • Factor pairs: 692 = 1 x 692, 2 x 346, or 4 x 173
  • Taking the factor pair with the largest square number factor, we get √692 = (√4)(√173) = 2√173 ≈ 26.30589.

Today is Thanksgiving in the United States. Regardless of where we live, there are reasons to be grateful today and every other day. Corrie ten Boom was grateful even when her circumstances were horrible. Here’s a Thanksgiving-themed puzzle to solve:

692 Turkey Puzzle

Sometimes color in a puzzle is a distraction. Here’s the same puzzle minus the color:

692 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-23

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Here’s a little more about the number 692:

Because 173 is one of its factors, 692 is the hypotenuse of Pythagorean triple 208-660-692. What is the greatest common factor of those three numbers?

692 is a palindrome in 2 bases:

  • 221122 BASE 3; note that 2(243) + 2(81) + 1(27) + 1(9) + 2(3) + 2(1) = 692
  • 848 BASE 9; note that 8(81) + 4(9) + 8(1) = 692

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692 Logic

Consecutive Numbers That Add up to 684

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684 is made from three consecutive even numbers so it is divisible by 3. The number in the middle of those consecutive numbers is divisible by 3 so 684 is also divisible by 9.

684 can be expressed as the sum of consecutive numbers in several ways:

  • 227 + 228 + 229 = 684 because one of its factor pairs is 3 x 228.
  • 72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80 = 684 because one of its factor pairs is 9 x 76.
  • 82 + 83 + 84 + 85 + 86 + 87 + 88 + 89 = 684 because it is greater than 32, is divisible by 4 but not by 8, and 684/8 = 85 1/2.
  • 27  + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 = 684 because one of its factor pairs is 19 x 36.
  • 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 = 684 because it is greater than 288, is divisible by 12 but not by 24, and 684/24 = 28 1/2.

684 is the sum of consecutive prime numbers 337 and 347.

684 is also the sum of the sixteen prime numbers from 13 to 73. Can you list all those prime numbers?

Best of all, the cubes of consecutive numbers 5, 6, and 7 add up to 684.

684 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-16

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  • 684 is a composite number.
  • Prime factorization: 684 = 2 x 2 x 3 x 3 x 19, which can be written 684 = (2^2) x (3^2) x 19
  • 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 684 has exactly 18 factors.
  • Factors of 684: 1, 2, 3, 4, 6, 9, 12, 18, 19, 36, 38, 57, 76, 114, 171, 228, 342, 684
  • Factor pairs: 684 = 1 x 684, 2 x 342, 3 x 228, 4 x 171, 6 x 114, 9 x 76, 12 x 57, 18 x 38 or 19 x 36
  • Taking the factor pair with the largest square number factor, we get √684 = (√36)(√19) = 6√19 ≈ 26.15339.

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684 Logic

 

677 and Level 4

/ ivasallay / Leave a comment

677 = 26² + 1², and it is the hypotenuse of the primitive Pythagorean triple 52-675-677 which was calculated using 2(26)(1),  26² – 1²,  26² + 1².

677 is also the sum of the eleven prime numbers from 41 to 83.

677 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-11-09

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  • 677 is a prime number.
  • Prime factorization: 677 is prime.
  • The exponent of prime number 677 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 677 has exactly 2 factors.
  • Factors of 677: 1, 677
  • Factor pairs: 677 = 1 x 677
  • 677 has no square factors that allow its square root to be simplified. √677 ≈ 26.01922.

How do we know that 677 is a prime number? If 677 were not a prime number, then it would be divisible by at least one prime number less than or equal to √677 ≈ 26.0. Since 677 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 677 is a prime number.

Here’s another way we know that 677 is a prime number: Since 677 ÷ 4 has a remainder of one, and 26² + 1² = 677, and 126 and 1 have no common prime factors, 677 will be prime unless it is divisible by a primitive Pythagorean hypotenuse less than or equal to √677 ≈ 26.0. Since 667 is not divisible by 5, 13, or 17, we know that 677 is a prime number.

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677 Logic

669 and Level 4

/ ivasallay / Leave a comment

Since all its digits are divisible by 3, obviously 669 is divisible by 3.

222 + 223 + 224 = 669. Thus 669 is the sum of 3 consecutive numbers. Prime factor 223 is the middle number in the sum.

Also consecutive numbers 109 + 110 + 111 + 112 + 113 + 114 = 669

669 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-11-02

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  • 669 is a composite number.
  • Prime factorization: 669 = 3 x 223
  • 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 669 has exactly 4 factors.
  • Factors of 669: 1, 3, 223, 669
  • Factor pairs: 669 = 1 x 669 or 3 x 223
  • 669 has no square factors that allow its square root to be simplified. √669 ≈ 25.865034.

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669 Logic

Ricardo shows the solution here:

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662 More Candy Corn

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662 is the sum of the twelve prime numbers from 31 to 79.

662 Puzzle Candy Corn

Print the puzzles or type the solution on this excel file: 10 Factors 2015-10-26

Here’s the same candy corn puzzle but less colorful.

662 Puzzle

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  • 662 is a composite number.
  • Prime factorization: 662 = 2 x 331
  • 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 662 has exactly 4 factors.
  • Factors of 662: 1, 2, 331, 662
  • Factor pairs: 662 = 1 x 662 or 2 x 331
  • 662 has no square factors that allow its square root to be simplified. √662 ≈ 25.72936.

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662 Logic

Ricardo tweeted the solution to the puzzle so I’m including it here as well.

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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

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  • 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.

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653 Logic

648 and Level 5

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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

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  • 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…

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648 Logic