A Multiplication Based Logic Puzzle

Simplifying √726

  • 726 is a composite number.
  • Prime factorization: 726 = 2 x 3 x 11 x 11, which can be written 726 = 2 x 3 x (11^2)
  • The exponents in the prime factorization are 1, 1, and 2. Adding one to each and multiplying we get (1 + 1)(1 + 1)(2 + 1) = 2 x 2 x 3 = 12. Therefore 726 has exactly 12 factors.
  • Factors of 726: 1, 2, 3, 6, 11, 22, 33, 66, 121, 242, 363, 726
  • Factor pairs: 726 = 1 x 726, 2 x 363, 3 x 242, 6 x 121, 11 x 66, or 22 x 33
  • Taking the factor pair with the largest square number factor, we get √726 = (√121)(√6) = 11√6 ≈ 26.944387.

Most numbers whose square roots can be simplified are divisible by 4, 9 or both. If I were trying to simplify √726, I would first apply divisibility tricks for 4 and 9 to the number 726:

  • 726 is even so it is divisible by 2, but since 26 is not divisible by 4, neither is 726.
  • Also notice that 7 + 2 + 6 = 15, a number divisible by 3, but not by 9. That means 726 is divisible by 3, but not by 9.

Thus 726 is NOT divisible by either 4 or 9. I will make a little cake to see if it is divisible by any other square numbers.

Dividing by 2 and then by 3 could be a good place to start, but rather than do those two divisions, I would simply do one division as I divide 726 by 6:

I would immediately recognize that 121 is 11 × 11, so I don’t need to make any more layers for this factoring cake!

To simplify √726, take the square root of everything on the outside of this cake (the numbers in red): Thus, √726 = (√6)(√121) = 11√6.

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This level 3 puzzle is not difficult to do. If you start at the top of the first column and work down cell by cell, you’ll catch the rhythm of finding all the factors that can make this puzzle function as a multiplication table:

726 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2015-12-28

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Here are a few more facts about the number 726:

726 is the average of 11^2 and 11^3 which is a characteristic that also makes it the 11th Pentagonal pyramidal number.

{724, 725, 726} is only the second set of three consecutive numbers in which each number in the set is the product of exactly one number squared and one different number. {603, 604, 605} was the first set.

726 is the sum of consecutive prime numbers 359 and 367.

726 is a palindrome in three bases:

  • 10401 BASE 5; note that 1(625) + 0(125) + 4(25) + 0(5) + 1(1) = 726.
  • 141 BASE 25; note that 1(625) + 4(25) + 1(1) = 726.
  • MM BASE 32 (M = 22 base 10); note that 22(32) + 22(1) = 726.

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726 Factors

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Comments on: "Simplifying √726" (4)

  1. “726 is a palindrome in three bases:” Reading this makes me happy.

    Liked by 1 person

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