A Multiplication Based Logic Puzzle

Posts tagged ‘Factors’

882 Factor Trees for the First Day of Autumn

September 22 was the first day of autumn. Leaves are already beginning to fall from the trees.

To rake up the leaves for 882, you might first notice that it’s even. The logical thing to do would be to first divide 882 by two. . . But perhaps you might notice that 8 + 8 + 2 = 18, a number divisible by nine, so you might just as logically begin by dividing 882 by 9. The first step you take determines how the factor tree looks.

882 has many possible factor trees but these two are probably the most common.

You can rake the leaves up this way or you can rake them up that way, but when you rake up the leaves from 882’s factor trees, you always get the same prime factors: 2, 3, 3, 7, and 7.

Here’s a little more about the number 882:

882 has eighteen factors. The greatest number less than 882 with eighteen factors is 828. Now get this: 288 also has eighteen factors. That means that every possible combination of 8-8-2 has exactly eighteen factors!

882 has some interesting representations in some other bases:

  • 616 BASE 12, because 6(12²) + 1(12)¹ + 6(12º) = 882
  • 242 BASE 20, because 2(20²) + 4(20)¹ + 2(20º) = 882
  • 200 BASE 21, because 2(21²) = 882

882 is also the sum of consecutive primes: 439 + 443 = 882

  • 882 is a composite number.
  • Prime factorization: 882 = 2 × 3 × 3 × 7 × 7, which can be written 882 = 2 × 3² × 7²
  • The exponents in the prime factorization are 1, 2 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1)(2 + 1) = 2 × 3 × 3 = 18. Therefore 882 has exactly 18 factors.
  • Factors of 882: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 49, 63, 98, 126, 147, 294, 441, 882
  • Factor pairs: 882 = 1 × 882, 2 × 441, 3 × 294, 6 × 147, 7 × 126, 9 × 98, 14 × 63, 18 × 49 or 21 × 42
  • Taking the factor pair with the largest square number factor, we get √882 = (√441)(√2) = 21√2 ≈ 29.6984848.

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881 What Level Should This Puzzle Be?

Before I started this blog I shared a sheet of six puzzles with a coworker. The most difficult puzzle on the sheet looked similar to this one.

Print the puzzles or type the solution on this excel file: 10-factors-875-885

He skipped ALL the easier puzzles and went straight for the most difficult one. Even though I advised him to use logic to solve the puzzle, he used guess and check and solved the puzzle within a couple of minutes. He then bragged that he could also solve a difficult Sudoku puzzle in about five minutes. He told me that my puzzles could never be a challenge to him and he wasn’t interested in ever doing another one. Ouch.

After that experience, when I began publishing my puzzles on my blog, I made sure the most difficult puzzle on the sheet were more difficult than his puzzle was.

Still I like this level of puzzle. I’m just not sure where it should be categorized.

It’s more difficult than level 4 because 20 and 16 have more than one possible common factor. However, 20 and 16 are the only set of multiple clues in any row or column, so it’s easier than a level 6. It doesn’t exactly qualify as a level 5 so I’m not assigning it that level.

Logic is still very important in finding the solution, although I suppose some lucky guess-and-checker might find it without logic. I think most people would only get frustrated if they just guessed and checked.

So give this puzzle a try. I’m calling it level ????, and its difficulty level is somewhere between a level 4 and a level 6.

Here are a few facts about the number 881:

25² + 16² = 881, so 881 is the hypotenuse of a Pythagorean triple which happens to be a primitive:

  • 369-800-881, which is calculated from 25² – 16², 2(25)(16), 25² + 16²

881 is the sum of the nine prime numbers from 79 to 113.

  • 881 is a prime number.
  • Prime factorization: 881 is prime.
  • The exponent of prime number 881 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 881 has exactly 2 factors.
  • Factors of 881: 1, 881
  • Factor pairs: 881 = 1 × 881
  • 881 has no square factors that allow its square root to be simplified. √881 ≈ 29.681644

How do we know that 881 is a prime number? If 881 were not a prime number, then it would be divisible by at least one prime number less than or equal to √881 ≈ 29.7. Since 881 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 881 is a prime number.

Here’s another way we know that 881 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 25² + 16² = 881 with 25 and 16 having no common prime factors, 881 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √881 ≈ 29.7. Since 881 is not divisible by 5, 13, 17, or 29, we know that 881 is a prime number.

 

879 and Level 4

879 consists of three consecutive numbers, 7, 8, 9, so it is divisible by 3.

879 is the hypotenuse of Pythagorean triple 204-855-879 which is 3 times (68-285-293)

Print the puzzles or type the solution on this excel file: 10-factors-875-885

  • 879 is a composite number.
  • Prime factorization: 879 = 3 × 293
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 879 has exactly 4 factors.
  • Factors of 879: 1, 3, 293, 879
  • Factor pairs: 879 = 1 × 879 or 3 × 293
  • 879 has no square factors that allow its square root to be simplified. √879 ≈ 29.64793

878 and Level 3

218 + 219 + 220 + 221 = 878; that’s the sum of four consecutive numbers.

438 + 440 = 878; that’s the sum of two consecutive even numbers.

878 is a palindrome in base 10 but not in any of the other bases from 2 to 36.

Print the puzzles or type the solution on this excel file: 10-factors-875-885

  • 878 is a composite number.
  • Prime factorization: 878 = 2 × 439
  • The exponents in the prime factorization are 1 and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1) = 2 × 2 = 4. Therefore 878 has exactly 4 factors.
  • Factors of 878: 1, 2, 439, 878
  • Factor pairs: 878 = 1 × 878 or 2 × 439
  • 878 has no square factors that allow its square root to be simplified. √878 ≈ 29.63106

877 and Level 2

29² + 6² = 877

That makes 877 the hypotenuse of a Primitive Pythagorean triple:

  • 348-805-877 calculated from 2(29)(6), 29² – 6², 29² + 6²

Print the puzzles or type the solution on this excel file: 10-factors-875-885

  • 877 is a prime number.
  • Prime factorization: 877 is prime.
  • The exponent of prime number 877 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 877 has exactly 2 factors.
  • Factors of 877: 1, 877
  • Factor pairs: 877 = 1 × 877
  • 877 has no square factors that allow its square root to be simplified. √877 ≈ 29.6141858

How do we know that 877 is a prime number? If 877 were not a prime number, then it would be divisible by at least one prime number less than or equal to √877 ≈ 29.6. Since 877 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 877 is a prime number.

Here’s another way we know that 877 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 29² + 6² = 877 with 29 and 6 having no common prime factors, 877 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √877 ≈ 29.6. Since 877 is not divisible by 5, 13, 17, or 29, we know that 877 is a prime number.

 

876 and Level 1

876 consists of three consecutive numbers 6, 7, 8, so 876 has to be divisible by 3. We can also conclude the following:

  • Since it’s even and divisible by 3, we know that 876 is also divisible by 6.
  • Since it is divisible by 3 and it’s last two digits are divisible by 4, we know that 876 is also divisible by 12.

Print the puzzles or type the solution on this excel file: 10-factors-875-885

876 is a palindrome in four other bases:

  • 727 BASE 11, because 7(121) + 2(11) + 7(1) = 876
  • 525 BASE 13, because 5(13²) + 2(13¹) + 5(13º) = 876
  • 282 BASE 19, because 2(19²) + 8(19¹) + 2(19º) = 876
  • 1A1 BASE 25 (A is 10 base 10), because 1(25²) + 10(25¹) + 1(25º) = 876

876 is also the hypotenuse of Pythagorean triple, 576-660-876 which is 12 times (48-55-73).

  • 876 is a composite number.
  • Prime factorization: 876 = 2 × 2 × 3 × 73, which can be written 876 = 2² × 3 × 73
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 876 has exactly 12 factors.
  • Factors of 876: 1, 2, 3, 4, 6, 12, 73, 146, 219, 292, 438, 876
  • Factor pairs: 876 = 1 × 876, 2 × 438, 3 × 292, 4 × 219, 6 × 146, or 12 × 73,
  • Taking the factor pair with the largest square number factor, we get √876 = (√4)(√219) = 2√219 ≈ 29.597297

874 and Level 6

874 is the sum of the first 23 prime numbers:

  • 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 874

ALL of 874’s prime factors were included in that list.

Print the puzzles or type the solution on this excel file: 12 factors 864-874

  • 874 is a composite number.
  • Prime factorization: 874 = 2 × 19 × 23
  • 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 874 has exactly 8 factors.
  • Factors of 874: 1, 2, 19, 23, 38, 46, 437, 874
  • Factor pairs: 874 = 1 × 874, 2 × 437, 19 × 46, or 23 × 38
  • 874 has no square factors that allow its square root to be simplified. √874 ≈ 29.56349.

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