A Multiplication Based Logic Puzzle

Posts tagged ‘Multiplication table’

710 and Level 1

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

  • 176 + 177 + 178 + 179 = 710; that’s four consecutive numbers.
  • 140 + 141 + 142 + 143 + 144 = 710; that’s five consecutive numbers.
  • 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 = 710; that’s 20 consecutive numbers.

710 is the sum of the twenty prime numbers from 3 to 73.

Because 5 is one of its factors, 710 is the hypotenuse of Pythagorean triple 426-568-710. What is the greatest common factor of those three numbers?

710 is palindrome 868 in BASE 9; note that 8(81) + 6(9) + 8(1) = 710.

710 Puzzle

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

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  • 710 is a composite number.
  • Prime factorization: 710 = 2 x 5 x 71
  • 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 710 has exactly 8 factors.
  • Factors of 710: 1, 2, 5, 10, 71, 142, 355, 710
  • Factor pairs: 710 = 1 x 710, 2 x 355, 5 x 142, or 10 x 71
  • 710 has no square factors that allow its square root to be simplified. √710 ≈ 26.645825.

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710 Factors

705 and Level 2

Because 5 is one of its factors, 705 is the hypotenuse of the Pythagorean triple 423-564-705. What is the greatest common factor of those three numbers?

705 is the sum of consecutive numbers several different ways:

  • 352 + 353 = 705; (2 consecutive numbers)
  • 234 + 235 + 236 = 705; (3 consecutive numbers)
  • 139 + 140 + 141 + 142 + 143 = 705; (5 consecutive numbers)
  • 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 + 54 = 705; (15 consecutive numbers)

705 is palindrome 1A1 in base 22; note that A is equivalent to 10 base 10, 22² = 484, and 1(484) + 10(22) + 1(1) = 705.

705 Puzzle

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

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  • 705 is a composite number.
  • Prime factorization: 705 = 3 x 5 x 47
  • 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 705 has exactly 8 factors.
  • Factors of 705: 1, 3, 5, 15, 47, 141, 235, 705
  • Factor pairs: 705 = 1 x 705, 3 x 235, 5 x 141, or 15 x 47
  • 705 has no square factors that allow its square root to be simplified. √705 ≈ 26.551836.

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705 Factors

704 and Level 1

7 – 0 + 4 = 11 so 704 is divisible by 11.

Here’s a fun multiplication fact: 8 x 88 = 704

Also 704 is a palindrome in two different bases:

  • 1I1 in BASE 19; note 19² = 361, I is equivalent to 18 base 10, and 1(361) + 18(19) + 1(1) = 704
  • MM in BASE 31; note M is equivalent to 22 base 10, and 22(31) = 22(1) = 704

704 Puzzle

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

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  • 704 is a composite number.
  • Prime factorization: 704 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 11, which can be written 704 = (2^6) ⋅ 11
  • The exponents in the prime factorization are 6, and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1) = 7 ⋅ 2 = 14. Therefore 704 has exactly 14 factors.
  • Factors of 704: 1, 2, 4, 8, 11, 16, 22, 32, 44, 64, 88, 176, 352, 704
  • Factor pairs: 704 = 1 x 704, 2 x 352, 4 x 176, 8 x 88, 11 x 64, 16 x 44, or 22 x 32
  • Taking the factor pair with the largest square number factor, we get √704 = (√64)(√11) = 8√11 ≈ 26.532998.

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704 Factors

695 and Level 1

695 is the sum of the eighteen prime numbers from 7 to 73.

695 is also the hypotenuse of the Pythagorean triple 417-556-695. What is the greatest common factor of those three numbers?

137 + 138 + 139 + 140 + 141 = 695 so it is the sum of 5 consecutive numbers.

Since 695 is odd, it is the sum of two consecutive numbers, 347 and 348.

695 Puzzle

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

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  • 695 is a composite number.
  • Prime factorization: 695 = 5 x 139
  • 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 695 has exactly 4 factors.
  • Factors of 695: 1, 5, 139, 695
  • Factor pairs: 695 = 1 x 695 or 5 x 139
  • 695 has no square factors that allow its square root to be simplified. √695 ≈ 26.36285.

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695 Factors

691 and Level 3

691 is the sum of the thirteen prime numbers from 29 to 79.

691 is equivalent to palindrome 171 in BASE 23. Note that 1(529) + 7(23) + 1(1) = 691.

691 Puzzle

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

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  • 691 is a prime number.
  • Prime factorization: 691 is prime and cannot be factored.
  • The exponent of prime number 691 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 691 has exactly 2 factors.
  • Factors of 691: 1, 691
  • Factor pairs: 691 = 1 x 691
  • 691 has no square factors that allow its square root to be simplified. √691 ≈ 26.2868789.

How do we know that 691 is a prime number? If 691 were not a prime number, then it would be divisible by at least one prime number less than or equal to √691 ≈ 26.3. Since 691 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 691 is a prime number.

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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 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.

691 Factors

687 and Level 6

687 is made from three consecutive numbers so it is divisible by 3.

687 is the hypotenuse of Pythagorean triple 180-663-687. What is the greatest common factor of those three numbers?

687 Puzzle

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

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  • 687 is a composite number.
  • Prime factorization: 687 = 3 x 229
  • 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 687 has exactly 4 factors.
  • Factors of 687: 1, 3, 229, 687
  • Factor pairs: 687 = 1 x 687 or 3 x 229
  • 687 has no square factors that allow its square root to be simplified. √687 ≈ 26.21068.

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687 Logic

Ricardo tweeted the solution for this one:

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670 and Level 5

Because 5 is one of its factors, 670 is the hypotenuse of the Pythagorean triple 402-536-670. Which factor of 670 is the greatest common factor of those three numbers?

670 Puzzle

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

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  • 670 is a composite number.
  • Prime factorization: 670 = 2 x 5 x 67
  • 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 670 has exactly 8 factors.
  • Factors of 670: 1, 2, 5, 10, 67, 134, 335, 670
  • Factor pairs: 670 = 1 x 670, 2 x 335, 5 x 134, or 10 x 67
  • 670 has no square factors that allow its square root to be simplified. √670 ≈ 25.884358.

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670 Logic

Thank you, Ricardo, for tweeting the solution:

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