662 More Candy Corn

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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652 and Level 3

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.

Stetson.edu 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

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  • 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.

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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

649 and Level 6

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

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  • 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.

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649 Logic

648 and Level 5

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. Stetson.edu 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

639 and Level 4

639 is the sum of the 20 prime numbers from 2 to 71, its largest prime factor.

639 is made from 3 numbers that are divisible by 3 so 639 is divisible by 3 AND by 9.

Puzzle #639 has a tricky clue in it, but I’m sure you can still solve it.

639 Puzzle

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

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  • 639 is a composite number.
  • Prime factorization: 639 = 3 x 3 x 71, which can be written 639 = (3^2) x 71
  • 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 639 has exactly 6 factors.
  • Factors of 639: 1, 3, 9, 71, 213, 639
  • Factor pairs: 639 = 1 x 639, 3 x 213, or 9 x 71
  • Taking the factor pair with the largest square number factor, we get √639 = (√9)(√71) = 3√71 ≈ 25.278449.

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639 Logic

637 and Level 2

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

633 and Level 5

  • 633 = 210 + 211 + 212 (three consecutive numbers)
  • 633 = 209 + 211 + 213 (three consecutive odd numbers)
  • 633 = 199 + 211 + 223 (three consecutive prime numbers)

Each of the sums above has 3 numbers, and 3 is a prime factor of 633. The middle number in each of the sums is 211 which is the other prime factor of 633.

Each of its two prime factors is 104 away from their average, 107.

Thus (107^2) – (104^2) = 633

The numbers in 633’s other factor pair are 1 and 633, and they are each 316 away from their average, 317.

Thus (317^2) – (316^2) = 633

633 Puzzle

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

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  • 633 is a composite number.
  • Prime factorization: 633 = 3 x 211
  • 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 633 has exactly 4 factors.
  • Factors of 633: 1, 3, 211, 633
  • Factor pairs: 633 = 1 x 633 or 3 x 211
  • 633 has no square factors that allow its square root to be simplified. √633 ≈ 25.15949.

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633 Logic

 

 

632 and Level 4

Since 632 is greater than 120 and is divisible by 8 but not by 16, it can be written as the sum of 16 consecutive counting numbers:

32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 = 632

632 Puzzle

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

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  • 632 is a composite number.
  • Prime factorization: 632 = 2 x 2 x 2 x 79, which can be written 632 = (2^3) x 79
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 x 2 = 8. Therefore 632 has exactly 8 factors.
  • Factors of 632: 1, 2, 4, 8, 79, 158, 316, 632
  • Factor pairs: 632 = 1 x 632, 2 x 316, 4 x 158, or 8 x 79
  • Taking the factor pair with the largest square number factor, we get √632 = (√4)(√158) = 2√158 ≈ 25.13961.

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632 Logic

629 and Level 1

629 is the sum of the 17 prime numbers from 7 to 71. Both of its prime factors, 17 and 37, are included in that list.

17 and 37 are both 10 numbers away from their average, 27. That means that 629 + 10² = 729 or 27².

25² + 2² = 629 and 23² + 10² = 629. Notice that 629 plus or minus 100 is a square number.

Both of 629’s prime factors have a remainder of one when divided by four so 629 is the hypotenuse of four Pythagorean triples, two of which are primitives.

  • 100-621-629, a primitive that reminds me of another primitive, 20-21-29
  • 204-595-629, three numbers whose greatest common factor is 17
  • 296-555-629, three numbers whose greatest common factor is 37
  • 429-460-629, a primitive whose shorter leg is exactly 200 less than its hypotenuse.

629 Puzzle

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

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  • 629 is a composite number.
  • Prime factorization: 629 = 17 x 37
  • 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 629 has exactly 4 factors.
  • Factors of 629: 1, 17, 37, 629
  • Factor pairs: 629 = 1 x 629 or 17 x 37
  • 629 has no square factors that allow its square root to be simplified. √629 ≈ 25.079872.

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629 Factors

616 and Level 2

The Padovan sequence produces a lovely spiral of equilateral triangles similar to the spiral made from golden rectangles and the Fibonacci sequence.

The Padovan sequence begins with the following numbers: 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86, 114, 151, 200, 265, 351, 465, 616, 816, 1081 …

The first two numbers in the Fibonacci sequence are both 1’s. After that a number, n, in the Fibonacci sequence is found by adding together (n-2) and (n-1).

The first three numbers in the Padovan sequence are all 1’s. After that a number, n, in the Padovan sequence is found by adding together (n-3) and (n-2), and 616 is one of those numbers.

616 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-09-14

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  • 616 is a composite number.
  • Prime factorization: 616 = 2 x 2 x 2 x 7 x 11, which can be written 616 = (2^3) x 7 x 11
  • The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 x 2 x 2 = 16. Therefore 616 has exactly 16 factors.
  • Factors of 616: 1, 2, 4, 7, 8, 11, 14, 22, 28, 44, 56, 77, 88, 154, 308, 616
  • Factor pairs: 616 = 1 x 616, 2 x 308, 4 x 154, 7 x 88, 8 x 77, 11 x 56, 14 x 44, or 22 x 28
  • Taking the factor pair with the largest square number factor, we get √616 = (√4)(√154) = 2√154 ≈ 24.819347.

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616 Factors