A Multiplication Based Logic Puzzle

Posts tagged ‘triangular’

What Kind of Shape Is 946 In?

First of all, 946 is the sum of the numbers from 1 to 43, so it is the 43rd triangular number.

Every other triangular number is also a hexagonal number. Since 946 is the 43rd triangular number, and 43 is an odd number, 946 is also the 22nd hexagonal number. 946 is the 22nd hexagonal number because 22(2(22) – 1) = 22(43) = 946.

But that’s not all. 946 is different than any previous hexagonal number. 946 is the smallest hexagonal number that is also a hexagonal pyramidal number. It is, in fact, the 11th hexagonal pyramidal number. That means if you stack the hexagons in the graphic below in order from largest to smallest, you would get a hexagonal pyramid made with 946 tiny squares. That’s pretty cool, I think.

 

467 + 479 = 946 so 946 is the sum of two consecutive prime numbers.
946 is also the sum of the twenty prime numbers from 11 to 89.

946 is palindrome 181 in BASE 27 because
1(27²) + 8(27¹) + 1(27⁰) = 729 + 216 + 1 = 946

  • 946 is a composite number.
  • Prime factorization: 946 = 2 × 11 × 43
  • 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 × 2 × 2 = 8. Therefore 946 has exactly 8 factors.
  • Factors of 946: 1, 2, 11, 22, 43, 86, 473, 946
  • Factor pairs: 946 = 1 × 946, 2 × 473, 11 × 86, or 22 × 43
  • 946 has no square factors that allow its square root to be simplified. √946 ≈ 30.75711

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741 and Level 3

Look at 741’s three digits: 4 = (1/2)(7 + 1) so 741 is divisible by 3.

  • 741 is a composite number.
  • Prime factorization: 741 = 3 x 13 x 19
  • 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 741 has exactly 8 factors.
  • Factors of 741: 1, 3, 13, 19, 39, 57, 247, 741
  • Factor pairs: 741 = 1 x 741, 3 x 247, 13 x 57, or 19 x 39
  • 741 has no square factors that allow its square root to be simplified. √741 ≈ 27.221315.

741-factor-pairs

Here is today’s puzzle:

741 Puzzle

Print the puzzles or type the solution on this excel file: 12 Factors 2016-01-11

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Here are more 741 number facts:

19 x 39 = 741. Since 19 and 39 are both exactly 10 away from their average, 29, we know that 741 is exactly 100 away from 29² = 841.

19 x 39 = (38/2 )(38 + 1) so 741 is the 38th triangular number.

741 can be expressed as the sum of consecutive numbers seven ways:

  • 370 + 371 = 741; that’s 2 consecutive numbers.
  • 246 + 247 + 248 = 741; that’s 3 consecutive numbers.
  • 121 + 122 + 123 + 124 + 125 + 126 = 741; that’s 6 consecutive numbers.
  • 51 + 52 + 53 + 54 + 55 + 56 + 57 + 58 + 59 + 60 + 61 + 62 + 63 = 741; that’s 13 consecutive numbers.
  • 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 = 741; that’s 19 consecutive numbers.
  • 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 = 741; that’s 26 consecutive numbers.
  • 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 = 741; that’s 38 consecutive numbers confirming that 741 is the 38th triangular number.

Since 13 is one of its factors, 741 is the hypotenuse of the Pythagorean triple 285-684-741. The greatest common factor of those 3 numbers can be found in the same factor pair as 13.

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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.

741 Factors

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