Level 2 Puzzle

696 There are lots of goodies in this Christmas Stocking

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  • 696 is a composite number.
  • Prime factorization: 696 = 2 x 2 x 2 x 3 x 29, which can be written 696 = (2^3) x 3 x 29
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 696 has exactly 16 factors.
  • Factors of 696: 1, 2, 3, 4, 6, 8, 12, 24, 29, 58, 87, 116, 174, 232, 348, 696
  • Factor pairs: 696 = 1 x 696, 2 x 348, 3 x 232, 4 x 174, 6 x 116, 8 x 87, 12 x 58, or 24 x 29
  • Taking the factor pair with the largest square number factor, we get √696 = (√4)(√174) = 2√174 ≈ 26.38181.

Today’s puzzle is meant to look like a Christmas stocking or boot that can be filled with lots of little treasures.

696 Puzzle

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

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What other facts did I find about the number 696?

696 is the sum of all the prime numbers from 71 to 103. Do you know what those eight prime numbers are?

696 is also the sum of consecutive odd numbers 347 and 349 which just happen to also be consecutive prime numbers.

Because 696 is a multiple of 29, it is the hypotenuse of Pythagorean triple 480-504-696. What is the greatest common factor of those three numbers?

696 is a palindrome in two different bases

  • 696 BASE 10; note that 6(100) + 9(10) + 6(1) = 696
  • OO BASE 28; note that O BASE 28 is equivalent to 24 in BASE 10, and  24(28) + 24(1) = 696

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696 Factors

 

When is 690 a Palindrome? In Base 16 and Base 29.

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  • 690 is a composite number.
  • Prime factorization: 690 = 2 x 3 x 5 x 23
  • 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 690 has exactly 16 factors.
  • Factors of 690: 1, 2, 3, 5, 6, 10, 15, 23, 30, 46, 69, 115, 138, 230, 345, 690
  • Factor pairs: 690 = 1 x 690, 2 x 345, 3 x 230, 5 x 138, 6 x 115, 10 x 69, 15 x 46, or 23 x 30
  • 690 has no square factors that allow its square root to be simplified. √690 ≈ 26.267851

Here is today’s factoring puzzle:

 

690 Puzzle

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

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Here is a little more about the number 690:

690 is the sum of the six prime numbers from 103 to 131. Do you know what all of those prime numbers are?

690 is also the hypotenuse of Pythagorean triple 414-552-690. What is the greatest common factor of those three numbers?

In BASE 10 we use the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. For bases smaller than 10 we use only the digits less than that base. For bases greater than 10, we might use A to represent 10, B to represent 11, and so forth all the way to Z representing 35 in BASE 36.

690 is a palindrome in two bases that require us to use letters of the alphabet to represent it:

  • 2B2 in BASE 16; note that 2(256) + 11(16) + 2(1) = 690
  • NN in BASE 29; note that 23(29) + 23(1) = 690. (N is the 14th letter of the alphabet and 14 + 9 = 23)

NN looks like it is divisible by 11, but remember that 11 base 29 is the same as 30 in base 10.

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690 Factors

682 Deserves a Lot of Exclamation Points!!!

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682 is the sum of the four prime numbers from 163 to 179. 682 is also the sum of the ten prime numbers from 47 to 89.

6 – 8 + 2 = 0 so 682 is divisible by 11.

OEIS.org shared another amazing relationship between the number 682 and the number 11:

682 factorials

Besides the obvious inclusion of the digits 6-8-2, notice in the factorial expression that 11 is in each numerator and that 11 is also the sum of the numbers in each denominator.

682 Puzzle

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

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  • 682 is a composite number.
  • Prime factorization: 682 = 2 x 11 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 682 has exactly 8 factors.
  • Factors of 682: 1, 2, 11, 22, 31, 62, 341, 682
  • Factor pairs: 682 = 1 x 682, 2 x 341, 11 x 62, or 22 x 31
  • 682 has no square factors that allow its square root to be simplified. √682 ≈ 26.1151297.

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682 Factors

675 and Level 2

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675 is made from 3 consecutive numbers so 675 is divisible by 3. Since the middle number of those consecutive numbers, 6, is divisible by 3, we know that 675 is also divisible by 9.

Since the last two digits of 675 is a multiple of 25, we know that 675 is divisible by 25.

Since 5 and 25 are two of its factors, 675 is the hypotenuse of two Pythagorean triples: 405-540-675 and 189-648-675. What is the greatest common factor of each set of three numbers?

The numbers in one of 675’s factor pairs, 25 and 27, are each exactly one number away from 26, their average. That means we are just one number away from 26².

Thus, 675 equals 26² – 1² which can be factored into (26 + 1)(26 – 1) so (26 + 1)(26 – 1) = 27 x 25 = 675.

675 Puzzle

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

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  • 675 is a composite number.
  • Prime factorization: 675 = 3 x 3 x 3 x 5 x 5, which can be written 675 = (3^3) x (5^2)
  • 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 675 has exactly 12 factors.
  • Factors of 675: 1, 3, 5, 9, 15, 25, 27, 45, 75, 135, 225, 675
  • Factor pairs: 675 = 1 x 675, 3 x 225, 5 x 135, 9 x 75, 15 x 45, or 25 x 27
  • Taking the factor pair with the largest square number factor, we get √675 = (√225)(√3) = 15√3 ≈ 25.98076.

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675 Factors

667 and Level 2

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Since 29 is one of its factor, 667 is the hypotenuse of the Pythagorean triple 460-483-667. What is the greatest common factor of those three numbers?

The numbers in the factor pair 23 and 29 are each exactly three numbers away from 26, their average. That means we are just 3² numbers away from 26².

Indeed, 667 equals 26² – 3² which can be factored into (26 + 3)(26 – 3) so (26 + 3)(26 – 3) = 29 x 23 = 667.

667 Puzzle

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

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  • 667 is a composite number.
  • Prime factorization: 667 = 23 x 29
  • 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 667 has exactly 4 factors.
  • Factors of 667: 1, 23, 29, 667
  • Factor pairs: 667 = 1 x 667 or 23 x 29
  • 667 has no square factors that allow its square root to be simplified. √667 ≈ 25.826343.

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667 Factors

659 Jack O’lantern Puzzle

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659 is the sum of the 7 prime numbers from 79 to 107. Can you list them all before you add them up?

Enjoy this Jack O’lantern puzzle.

659 Puzzle

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

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  • 659 is a prime number.
  • Prime factorization: 659 is prime.
  • The exponent of prime number 659 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 659 has exactly 2 factors.
  • Factors of 659: 1, 659
  • Factor pairs: 659 = 1 x 659
  • 659 has no square factors that allow its square root to be simplified. √659 ≈ 25.670995.

How do we know that 659 is a prime number? If 659 were not a prime number, then it would be divisible by at least one prime number less than or equal to √659 ≈ 25.7. Since 659 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 659 is a prime number.

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Since riccardo took the time to photograph the puzzle’s solution, I decided to include it in this post as well. Great job riccardo!

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651 and Level 2

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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

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  • 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.

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651 Factors

644 Divisibility Rules and Level 2

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Divisibility rules 1-11 applied to the number 644:

  1. All counting numbers are divisible by 1 so 644 IS divisible by 1.
  2. 644 is even so it IS divisible by 2.
  3. 6 + 4 + 4 = 14 which is not a multiple of 3 so 644 is NOT divisible by 3.
  4. The last two digits, 44, are divisible by 4, so 644 IS divisible by 4.
  5. The last digit is not 0 or 5, so 644 is NOT divisible by 5.
  6. 644 is divisible by 2, but not by 3 so 644 is NOT divisible by 6.
  7. Breaking off the last digit, doubling it and subtracting it from the remaining digits we get: 64 – 2(4) = 56, a multiple of 7, so 644 IS divisible by 7.
  8. 6 is an even digit, and 44 is divisible by 4 but not by 8 so 644 is NOT divisible by 8.
  9. 6 + 4 + 4 = 14 which is not a multiple of 9 so 644 is NOT divisible by 9.
  10. The last digit is not 0, so 644 is NOT divisible by 10.
  11. 6 – 4 + 4 = 6 which is not a multiple of 11 so 644 is NOT a multiple of 11.

Just for the fun of it let’s try a divisibility rule for 23. Break off the last digit, multiply it by 7 and add it to the remaining digits: 64 + 7(4) = 64 + 28 = 92.

Now apply the same rule to 92: 9 + 7(2) = 9 + 14 = 23, obviously a multiple of 23, so 92 and 644 are both divisible by 23.

644 Puzzle

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

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  • 644 is a composite number.
  • Prime factorization: 644 = 2 x 2 x 7 x 23, which can be written 644 = (2^2) x 7 x 23
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 644 has exactly 12 factors.
  • Factors of 644: 1, 2, 4, 7, 14, 23, 28, 46, 92, 161, 322, 644
  • Factor pairs: 644 = 1 x 644, 2 x 322, 4 x 161, 7 x 92, 14 x 46, or 23 x 28
  • Taking the factor pair with the largest square number factor, we get √644 = (√4)(√161) = 2√161 ≈ 25.377155

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644 Factors

637 and Level 2

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637 is the sum of two perfect squares, 441 and 196, so it is the hypotenuse of a Pythagorean triple, namely 245-588-637. The greatest common factor of those FIVE numbers is also a perfect square. What is it?

637 is the sum of the nineteen prime numbers from 3 to 71.

637 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-10-05

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  • 637 is a composite number.
  • Prime factorization: 637 = 7 x 7 x 13, which can be written 637 = (7^2) x 13
  • 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 637 has exactly 6 factors.
  • Factors of 637: 1, 7, 13, 49, 91, 637
  • Factor pairs: 637 = 1 x 637, 7 x 91, or 13 x 49
  • Taking the factor pair with the largest square number factor, we get √637 = (√49)(√13) = 7√13 ≈ 25.2388589.

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637 Factors

630 Factor Trees and Level 2

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Today’s Puzzle:

630 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-28

Factors for Today’s Puzzle:

630 Factors

Factor Trees for 630:

630 is the 7th number with exactly 24 factors. So far, the seven numbers counting numbers with 24 factors are 360, 420, 480, 504, 540, 600, and 630. No counting number less than 630 has more than 24 factors.

Two of those seven numbers make up the Pythagorean triple 378-504-630. Which factor of 630 is the greatest common factor of those three numbers in the triple?

Here are a few of the MANY possible factor trees for 630.

630 Factor Trees

Factors of 630:

  • 630 is a composite number.
  • Prime factorization: 630 = 2 x 3 x 3 x 5 x 7, which can be written 630 = 2 x (3^2) x 5 x 7
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 x 3 x 2 x 2 = 24. Therefore 630 has exactly 24 factors.
  • Factors of 630: 1, 2, 3, 5, 6, 7, 9, 10, 14, 15, 18, 21, 30, 35, 42, 45, 63, 70, 90, 105, 126, 210, 315, 630
  • Factor pairs: 630 = 1 x 630, 2 x 315, 3 x 210, 5 x 126, 6 x 105, 7 x 90, 9 x 70, 10 x 63, 14 x 45, 15 x 42, 18 x 35, or 21 x 30
  • Taking the factor pair with the largest square number factor, we get √630 = (√9)(√70) = 3√70 ≈ 25.09980.

Sum-Difference Puzzle:

630 has twelve factor pairs. One of the factor pairs adds up to 53, and a different one subtracts to 53. If you can identify those factor pairs, then you can solve this puzzle!

More about the Number 630:

630 is the sum of the six prime numbers from 97 to 113.

630 is the 35th triangular number because (35 x 36)/2 = 630. It is also the 18th hexagonal number because 18(2 x 18 – 1) = 630.

630 is a triangular number that is a multiple of other triangular numbers in more ways than you probably want to know:

  • 630 is three times the 20th triangular number, 210, because 3(20 x 21)/2 = 630.
  • 630 is 6 times the 14th triangular number, 105, because 6(14 x 15)/2 = 630.
  • 630 is 14 times the 9th triangular number, 45, because 14(9 x 10)/2 = 630.
  • 630 is 30 times the 6th triangular number, 21, because 30(6 x 7)/2 = 630.
  • 630 is 42 times the 5th triangular number, 15, because 42(5 x 6)/2 = 630.
  • 630 is 63 times the 4th triangular number, 10, because 63(4 x 5)/2 = 630.
  • 630 is 105 times the 3rd triangular number, 6, because 105(3 x 4)/2 = 630.
  • 630 is 210 times the 2nd triangular number, 3, because 210(2 x 3)/2 = 630.
  • and finally, 630 is 630 times the 1st triangular number, 1, because 630(1 x 2)/2 = 630.