Level 3 Puzzle

878 and Level 3

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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

869 and Level 3

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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

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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.

OEIS.org 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

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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√53 ≈ 29.1204396

837 and Level 3

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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

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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

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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

 

 

817 Tiny Squares

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Start at the top row of this level three puzzle and work down one cell at a time until you’ve written each number from 1 through 12 in both the top row and the first column. You will have solved the puzzle if all the clues given in the puzzle are the products of the numbers you wrote. You can do this!

Print the puzzles or type the solution on this excel file: 12 factors 815-820

  • 817 is a composite number.
  • Prime factorization: 817 = 19 x 43
  • 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 817 has exactly 4 factors.
  • Factors of 817: 1, 19, 43, 817
  • Factor pairs: 817 = 1 x 817 or 19 x 43
  • 817 has no square factors that allow its square root to be simplified. √817 ≈ 28.58321

817 is the sum of three consecutive prime numbers:

  • 269 + 271 + 277 = 817

Because it has two factor pairs in which the numbers in the pair are either both odd or both even, 817 can be written as the difference of two squares two different ways:

  • 817 = 31² – 12², which I’ve illustrated below.
  • 817 = 409² – 408²

Being able to be written as the difference of two squares means that 817 is a leg in two Pythagorean triples so:

  • 744² + 817² = 1105²
  • 817² + 333744² = 333745²

Last, but certainly not least, 817 is the difference of two consecutive cubes, namely 17³ – 16³ = 817. That means that 817 is the 17th centered hexagonal number as well! There are 17 squares on each side of the figure below and 817 tiny purple squares in all. The horizontal row in the middle has 2(17) – 1 = 33 tiny purple squares.

2(17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32) + 33 = 817.

That difference of two cubes also means that

  • 817 = 17³ – 16³ = (17 – 16)(17² + 16·17 + 16²) = (1)(289 + 272 + 256) = 817.

That’s from a³ – b³ = (a – b)(a² + ab + b²) and is illustrated below using 2D cubes instead of squares. In 3D, when the sides are folded up, the darker cubes can be claimed by two different sides so the figure will look like a 17 x 17 x 17 inside corner such as in a room where three perpendicular lines meet, but in 2D it’s just the flat net you see here:

797 and Mathematical Ways to Love

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Mathwithbaddrawings.com has some thoughtful and entertaining Ways to Tell a Mathematician that you love them.

Artful Maths wrote a post that includes beautiful mathematical origami valentines and a “string art” cardioid that is made with a pencil instead of string.

This puzzle could be another mathematical way to show some love:

797-love-puzzle

Print the puzzles or type the solution on this excel file: 12-factors-795-799

Here are a few facts about the number 797:

797 is a palindrome in three bases:

  • 797 BASE 10 because 7(100) + 9(10) + 7(1) = 797
  • 565 BASE 12 because 5(144) + 6(12) + 5(1) = 797
  • 494 BASE 13 because 4(169) + 9(13) + 4(1) = 797

But there’s one more palindromic fact about the number 797: It is the sum of two square numbers that are also palindromes!

  • 797 = 121 + 676. Note that 11² = 121 and 26² = 676.

Since it is the sum of two squares, 797 will also be the hypotenuse a Pythagorean triple:

  • 555 – 572 – 797 calculated from 26² – 11², 2(26)(11), 26² + 11².

797 is the sum of three squares seven different ways:

  • 28² + 3² + 2² = 797
  • 27² + 8² + 2² = 797
  • 24² + 14² + 5² = 797
  • 24² + 11² + 10² = 797
  • 22² + 13² + 12² = 797
  • 21² + 16² + 10² = 797
  • 20² + 19² + 6² = 797

797 is also the sum of the 15 prime numbers from 23 to 83:

  • 23 + 29 + 31 + 37+ 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 797

Here is the most basic information about the number 797:

  • 797 is a prime number.
  • Prime factorization: 797 is prime.
  • The exponent of prime number 797 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 797 has exactly 2 factors.
  • Factors of 797: 1, 797
  • Factor pairs: 797 = 1 x 797
  • 797 has no square factors that allow its square root to be simplified. √797 ≈ 28.231188.

How do we know that 797 is a prime number? If 797 were not a prime number, then it would be divisible by at least one prime number less than or equal to √797 ≈ 28.2. Since 797 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 797 is a prime number.

Here’s another way we know that 797 is a prime number: Since  its last two digits divided by 4 leave a remainder of 1, and 26² + 11² = 797 with 26 and 11 having no common prime factors, 797 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √797 ≈ 28.2. Since 797 is not divisible by 5, 13, or 17, we know that 797 is a prime number.

797-factor-pairs

 

790 and Level 3

/ ivasallay / Leave a comment
  • 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

Here is today’s puzzle:

790-puzzle

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

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Here’s a little more about the number 790:

There are four ways to make 790 using three squares:

  • 21² + 18² + 5² = 790
  • 27² + 6² + 5² = 790
  • 23² + 15² + 6² = 790
  • 22² + 15² + 9² = 790

790 is the hypotenuse of Pythagorean triple 474-632-790 which is 3-4-5 times 158.

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