A Multiplication Based Logic Puzzle

Posts tagged ‘pentagonal number’

715 and Level 5

  • 715 is a composite number.
  • Prime factorization: 715 = 5 x 11 x 13
  • 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 715 has exactly 8 factors.
  • Factors of 715: 1, 5, 11, 13, 55, 65, 143, 715
  • Factor pairs: 715 = 1 x 715, 5 x 143, 11 x 65, or 13 x 55
  • 715 has no square factors that allow its square root to be simplified. √715 ≈ 26.73948.

Can you solve today’s puzzle?

715 Puzzle

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

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Here are several more facts about the number 715:

715 is the 22nd pentagonal number because 22(3⋅22 – 1)/2 = 715.

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

  • 357 +358 = 715; that’s 2 consecutive numbers.
  • 141 + 142 + 143 + 144 + 145 = 715; that’s 5 consecutive numbers.
  • 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 = 715; that’s 11 consecutive numbers.
  • 49 + 50 + 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 = 715; that’s 13 consecutive numbers.

Since 5 and 13 are two of its factors, 715 is the hypotenuse of four Pythagorean triples.

  • 176-693-715
  • 275-660-715
  • 363-616-715
  • 429-572-715

Each of those triples has a greatest common factor. Can you find them all?

715 is also palindrome 1D1 in BASE 21 where D = 13 base 10; note that 1(21²) + 13(21) + 1(1) = 715.

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715 Logic

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532 and Level 3

532 is the sum of consecutive prime numbers 263 and 269.

532 is also the 19th pentagonal number.

532 Puzzle

Print the puzzles or type the solution on this excel file: 10 Factors 2015-06-22

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  • 532 is a composite number.
  • Prime factorization: 532 = 2 x 2 x 7 x 19, which can be written 532 = (2^2) x 7 x 19
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 x 2 x 2 = 12. Therefore 532 has exactly 12 factors.
  • Factors of 532: 1, 2, 4, 7, 14, 19, 28, 38, 76, 133, 266, 532
  • Factor pairs: 532 = 1 x 532, 2 x 266, 4 x 133, 7 x 76, 14 x 38, or 19 x 28
  • Taking the factor pair with the largest square number factor, we get √532 = (√4)(√133) = 2√133 ≈ 23.065125

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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. 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.

532 Factors

 

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