A Multiplication Based Logic Puzzle

Archive for the ‘Level 3 Puzzle’ Category

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

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869 and Level 3

8 – 6 + 9 = 11 so 869 is divisible by 11.

869 is equal to the sum of the 21 prime numbers from 5 to 83.

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

  • 869 is a composite number.
  • Prime factorization: 869 = 11 × 79
  • 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 869 has exactly 4 factors.
  • Factors of 869: 1, 11, 79, 869
  • Factor pairs: 869 = 1 × 869 or 11 × 79
  • 869 has no square factors that allow its square root to be simplified. √869 ≈ 29.4788

856 Rays of Light

When we learn something new, it is as if rays of light touch our minds. If you know how to multiply and divide, let this puzzle enlighten your mind. Just start at the top of the first column and work down cell by cell until the numbers 1 to 10 have been written in the first column and the top row and those corresponding numbers multiply together to give the clues in the puzzle.

Print the puzzles or type the solution on this excel file: 10-factors-853-863

 

  • 856 is a composite number.
  • Prime factorization: 856 = 2 × 2 × 2 × 107, which can be written 856 = 2³ × 107
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 856 has exactly 8 factors.
  • Factors of 856: 1, 2, 4, 8, 107, 214, 428, 856
  • Factor pairs: 856 = 1 × 856, 2 × 428, 4 × 214, or 8 × 107
  • Taking the factor pair with the largest square number factor, we get √856 = (√4)(√214) = 2√214 ≈ 29.2574777

Here are a few more advanced facts about the number 856:

856 is the 16th nonagonal number because 16(7⋅16-5)/2.

856 is the 19th centered pentagonal number because (5⋅19² + 5⋅19 + 2)/2 = 856.

Stetson.edu informs us that if the Fibonacci sequence didn’t start with 1, 1, but instead started with 1, 9, we would get 1, 9, 10, 19, 29, 48, 77, 125, 202, 327, 529, 856, …

848 and Level 3

Print the puzzles or type the solution on this excel file: 12 factors 843-852

848 is a palindrome, and all but three of its factors are palindromes, too. (Single digit numbers are also palindromes.)

848 is the sum of two squares: 28² + 8² = 848

848 is the hypotenuse of a Pythagorean triple:

  • 448-720-848, calculated from 2(28)(8), 28² – 8², 28² + 8²

844, 845, 846, 847, and 848 are the smallest five consecutive numbers whose square roots can be simplified.

  • 848 is a composite number.
  • Prime factorization: 848 = 2 × 2 × 2 × 2 × 53, which can be written 848 = 2⁴ × 53
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 848 has exactly 10 factors.
  • Factors of 848: 1, 2, 4, 8, 16, 53, 106, 212, 424, 848
  • Factor pairs: 848 = 1 × 848, 2 × 424, 4 × 212, 8 × 106, or 16 × 53
  • Taking the factor pair with the largest square number factor, we get √848 = (√16)(√53) = 4√13 ≈ 29.1204396

837 and Level 3

837 is repdigit RR in BASE 30 (R is 27 in base 10). All that means is that 27(30¹) + 27(30º) = 837, which naturally follows from the fact that 27(30 + 1) = 837.

837 has four odd factor pairs, so 837 can be written as the difference of two squares four different ways:

  • 837 × 1 = 837 means 419² – 418² = 837
  • 279 × 3 = 837 means 141² – 138² = 837
  • 93 × 9 = 837 means 51² – 42² = 837
  • 31 × 27 = 837 means 29² – 2² = 837

Hmm…837 is only four numbers away from the next perfect square, 841.

Print the puzzles or type the solution on this excel file: 10-factors-835-842

  • 837 is a composite number.
  • Prime factorization: 837 = 3 × 3 × 3 × 31, which can be written 837 = 3³ × 31
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 837 has exactly 8 factors.
  • Factors of 837: 1, 3, 9, 27, 31, 93, 279, 837
  • Factor pairs: 837 = 1 × 837, 3 × 279, 9 × 93, or 27 × 31
  • Taking the factor pair with the largest square number factor, we get √837 = (√9)(√93) = 3√93 ≈ 28.93095

 

 

831 and Level 3

Print the puzzles or type the solution on this excel file: 12 factors 829-834

Before I write a blog post, I look to see how the post number is expressed in different bases. Today I noticed that 831 is 30333 in BASE 4. I was intrigued by all those 3’s because I knew that 831 = 3 × 277. It seems logical that 277 would be 10111 in BASE 4, it turns out that it is! I looked at 831 in all the bases up to BASE 36. Did any others have only multiples of 3 as its digits? Yes, a few did, so I’ve made a chart of 277 and 831 in those five bases to make comparing them easy. I also used only base 10 numbers and not letters of the alphabet to represent the digits in the other bases. As you look at this chart, remember 3 × 277 = 831.

Why are those the ONLY bases for which 3 times the digits of 277 equals the digits for 831? Because in every other base, at least one of the digits times 3 will be greater than or equal to the base and some complicated carrying will have to take place to determine the digits for 831 in that base.

For example, 277 is palindrome 1 11 1 in BASE 12. Obviously 3 times 1 11 1 is 3 33 3. Since 33 is bigger than 12, we somehow end up with non-palindrome 5 9 3 in BASE 12 for 831. This is how that somehow happened: 33÷12 = 2R9. The 9 becomes the middle digit while the 2 is added to the original 3 to make the first digit, 5.

Here’s a little more about the number 831:

Because 277 is one of its factors, 831 is the hypotenuse of a Pythagorean triple: 345-756-831, which is 3 times primitive (115-252-277).

  • 831 is a composite number.
  • Prime factorization: 831 = 3 × 277
  • 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 831 has exactly 4 factors.
  • Factors of 831: 1, 3, 277, 831
  • Factor pairs: 831 = 1 × 831 or 3 × 277
  • 831 has no square factors that allow its square root to be simplified. √831 ≈ 8270706

 

 

824 and Level 3

824 is the sum of all the prime numbers from 61 all the way to 103, which just happens to be one of its prime factors!

Print the puzzles or type the solution on this excel file: 10-factors-822-828

824 is a leg in a few Pythagorean triples:

  • 618-824-1030 because that is 206 times (3-4-5)
  • 824-1545-1751 because that is 103 times (8-15-17)
  • 824-10593-10625 because 2(103)(4) = 824
  • 824-21210-21226 because 105² – 101² = 824
  • 824-42432-42440 because 2(206)(2) = 824
  • 824-84870-84874 because 207² – 205² = 824
  • Primitive 824-169743-169745 because 2(412)(1) = 824

Five of those triples were derived directly from 824’s factor pairs.

Two of the triples were derived indirectly:

  • What is (105+101)/2, (105-101)/2?
  • Also, what is (207+205)/2, (207+205)/2?

The answer to both questions is a factor pair of 824.

You can read more about finding Pythagorean triples for numbers that are divisible by 4 here.

  • 824 is a composite number.
  • Prime factorization: 824 = 2 × 2 × 2 × 103, which can be written 824 = 2³ × 103
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 824 has exactly 8 factors.
  • Factors of 824: 1, 2, 4, 8, 103, 206, 412, 824
  • Factor pairs: 824 = 1 × 824, 2 × 412, 4 × 206, or 8 × 103
  • Taking the factor pair with the largest square number factor, we get √824 = (√4)(√206) = 2√206 ≈ 28.7054

 

 

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