A Multiplication Based Logic Puzzle

Posts tagged ‘Prime factorization’

791 and Level 4

To see if 791 is divisible by 7, you could try either one of these divisibility tricks:

  • 791 is divisible by 7 because 79 – 2(1) = 77 which obviously is divisible by 7.
  • 791 is divisible by 7 because 79 + 5(1) = 84 which most people know is 12 × 7.

791-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

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  • 791 is a composite number.
  • Prime factorization: 791 = 7 x 113
  • 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 791 has exactly 4 factors.
  • Factors of 791: 1, 7, 113, 791
  • Factor pairs: 791 = 1 x 791 or 7 x 113
  • 791 has no square factors that allow its square root to be simplified. √791 ≈ 28.12472222.

791-factor-pairs

790 and Level 3

790’s prime factorization is 2 × 5 × 79. Since two of those numbers, 5 and 79, have a remainder of one when divided by four, I automatically know that 790 is the hypotenuse of exactly four Pythagorean triples.

790-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

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  • 790 is a composite number.
  • Prime factorization: 790 = 2 x 5 x 79
  • 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 790 has exactly 8 factors.
  • Factors of 790: 1, 2, 5, 10, 79, 158, 395, 790
  • Factor pairs: 790 = 1 x 790, 2 x 395, 5 x 158, or 10 x 79
  • 790 has no square factors that allow its square root to be simplified. √790 ≈ 28.106939.

790-factor-pairs

789 and Level 2

789 consists of exactly three consecutive numbers so it is divisible by 3.

789-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

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  • 789 is a composite number.
  • Prime factorization: 789 = 3 x 263
  • 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 789 has exactly 4 factors.
  • Factors of 789: 1, 3, 263, 789
  • Factor pairs: 789 = 1 x 789 or 3 x 263
  • 789 has no square factors that allow its square root to be simplified. √789 ≈ 28.08914.

789-factor-pairs

788 and Level 1

Since 88, its last two digits, are divisible by 4, we know that 788 and every other whole number ending in 88 is divisible by 4.

I learned the following fascinating fact about these six numbers starting with 788 from Stetson.edu:

788-consecutive-numbers

788 is also palindrome 404 in BASE 14. Note that 4(196) + 0(14) + 4(1) = 788.

788-puzzle

Print the puzzles or type the solution on this excel file: 10-factors-788-794

  • 788 is a composite number.
  • Prime factorization: 788 = 2 x 2 x 197, which can be written 788 = (2^2) x 197
  • 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 788 has exactly 6 factors.
  • Factors of 788: 1, 2, 4, 197, 394, 788
  • Factor pairs: 788 = 1 x 788, 2 x 394, or 4 x 197
  • Taking the factor pair with the largest square number factor, we get √788 = (√4)(√197) = 2√197 ≈ 28.071338.

788-factor-pairs

786 and Level 5

786 is even so it is divisible by 2. Also since 786 is made from 3 consecutive numbers, we can tell automatically that it is divisible by 3. Those two facts together mean 786 is also divisible by 6.

  • 786 is a composite number.
  • Prime factorization: 786 = 2 x 3 x 131
  • 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 786 has exactly 8 factors.
  • Factors of 786: 1, 2, 3, 6, 131, 262, 393, 786
  • Factor pairs: 786 = 1 x 786, 2 x 393, 3 x 262, or 6 x 131
  • 786 has no square factors that allow its square root to be simplified. √786 ≈ 28.03569.

786-factor-pairs

Today’s Find the Factors puzzle:

786-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Here’s a little more about the number 786:

786 is 123 in BASE 27 because 1(27²) + 2(27) + 3(1) = 786.

786 is the sum of two consecutive primes: 389 + 397 = 786

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786-logic

785 and Level 4

  • 785 is a composite number.
  • Prime factorization: 785 = 5 x 157
  • 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 785 has exactly 4 factors.
  • Factors of 785: 1, 5, 157, 785
  • Factor pairs: 785 = 1 x 785 or 5 x 157
  • 785 has no square factors that allow its square root to be simplified. √785 ≈ 28.01785.

785-factor-pairs

Now for today’s puzzle:

785-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Here’s a little more about the number 785:

785 is the sum of two squares two different ways:

  • 28² + 1² = 785
  • 23² + 16² = 785

785 is also the sum of three squares three different ways:

  • 26² + 10² + 3² = 785
  • 25² + 12² + 4² = 785
  • 19² + 18² + 10² = 785

Because its prime factorization is 5 × 157 (two numbers that are also the sum of two squares), 785 is the hypotenuse of four Pythagorean triples, two of which are primitive triples:

  • 56-783-785 primitive calculated from 2(28)(1), 28² – 1², 28² + 1²
  • 273-736-785 primitive calculated from 23² – 16², 2(23)(16), 23² + 16²
  • 425-660-785 which is 5 times 85-132-157
  • 471-628-785 which is 157 times 3-4-5

785 is also a palindrome in two different bases:

  • 555 BASE 12; note that 5(144) + 5(12) + 5(1) = 785
  • 101 BASE 28; note that 1(28²) + 0(28) + 1(1) = 785

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785-logic

 

784 and Level 3

If you know the multiplication facts up to 12 x 12, then it is obvious that 784 is divisible by 7. However, 784 is divisible by a whole lot more numbers than that. 784 can actually be evenly divided by 15 different numbers. Hmm, 15, that’s an odd number. A number’s factors always come in pairs. One of its factor pairs must contain the same factor twice, meaning 784 is a perfect square!

  • 784 is a composite number.
  • Prime factorization: 784 = 2 x 2 x 2 x 2 x 7 x 7, which can be written 784 = (2^4) x (7^2)
  • The exponents in the prime factorization are 4 and 2. Adding one to each and multiplying we get (4 + 1)(2 + 1) = 5 x 3 = 15. Therefore 784 has exactly 15 factors.
  • Factors of 784: 1, 2, 4, 7, 8, 14, 16, 28, 49, 56, 98, 112, 196, 392, 784
  • Factor pairs: 784 = 1 x 784, 2 x 392, 4 x 196, 7 x 112, 8 x 98, 14 x 56, 16 x 49, or 28 x 28
  • 784 is a perfect square. √784 = 28

784-factor-pairs

But that’s not the only thing remarkable about this perfect square: √784 is 28, the 7th triangular number, so like all other squared triangular numbers 784 has this additional property:

784-sum-of-consecutive-cubes

Just as 784 is a perfect square, five of the twelve clues in today’s puzzle are also perfect squares. But don’t let that fact trick you into writing the same factor in both the first column and the top row every time!

784-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-782-787

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Here’s more about the number 784:

784 can be written as the sum of three squares in exactly one way:

  • 24² + 12² + 8² = 784

784 is also a palindrome in Bases 13, 17, and 27:

  • 484 BASE 13; note that 4(169) + 8(13) + 4(1) = 784
  • 2C2 BASE 17 (C is 12 base 10); note that 2(289) + 12(17) + 2(1) = 784
  • 121 BASE 27; note that 1(27²) + 2(27) + 1(1) = 784

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784-factors

 

 

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