Level 2 Puzzle

916 Witch’s Hat

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Today’s puzzle looks a little like a witch’s hat. Solving it could be bewitching. It contains a lot of perfect squares, and it’s only a level 2. Give it a try! Just make sure you don’t write a number greater than ten in the top row or the first column. Use all the numbers from one to ten in both places. There is only one solution.

Print the puzzles or type the solution on this excel file: 10-factors-914-922

916 looks the same upside-down as it does right-side up, so it is a Strobogrammatic number.

It looks like someone put a spell on that number!

Not only that, in BASE 26, our 916 looks like 196. There is a whole lot of magic going on here!

30² + 4² = 916, so 916 is the hypotenuse of a Pythagorean triple:
240-884-916, which can be calculated from 2(30)(4), 30² – 4², 30² + 4².

  • 916 is a composite number.
  • Prime factorization: 916 = 2 × 2 × 229, which can be written 916 = 2² × 229
  • The exponents in the prime factorization are 2 and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 916 has exactly 6 factors.
  • Factors of 916: 1, 2, 4, 229, 458, 916
  • Factor pairs: 916 = 1 × 916, 2 × 458, or 4 × 229
  • Taking the factor pair with the largest square number factor, we get √916 = (√4)(√229) = 2√229 ≈ 30.2654919

 

 

906 and Level 2

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906 looks the same upside-down so it is a strobogrammatic number.

We can tell just by looking at 906 that it is divisible by 1, 2, and 3. That means it can also be evenly divided by 6.

906 is the sum of consecutive prime numbers: 449 + 457 = 906

It is also palindrome 636 in BASE 12 because 6(144) + 3(12) + 6(1) = 906.

Print the puzzles or type the solution on this excel file: 12 factors 905-913

  • 906 is a composite number.
  • Prime factorization: 906 = 2 × 3 × 151
  • 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 906 has exactly 8 factors.
  • Factors of 906: 1, 2, 3, 6, 151, 302, 453, 906
  • Factor pairs: 906 = 1 × 906, 2 × 453, 3 × 302, or 6 × 151
  • 906 has no square factors that allow its square root to be simplified. √906 ≈ 30.099833887

 

898 and Level 2

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27² + 13² = 898. That means that 898 is the hypotenuse of a Pythagorean triple:

  • 560-702-898, which is 2 times (280-351-449).

898 reads the same way frontwards and backwards so it is a palindrome in base 10.

It is also palindrome 747 in BASE 11 because 7(11²) + 4(11) + 7(1) = 898

AND it is palindrome 1G1 in BASE 23 (G is 16 in base 10) because 1(23²) + 16(23) +1(1) =898.

Print the puzzles or type the solution on this excel file: 10-factors-897-904

  • 898 is a composite number.
  • Prime factorization: 898 = 2 × 449
  • 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 898 has exactly 4 factors.
  • Factors of 898: 1, 2, 449, 898
  • Factor pairs: 898 = 1 × 898 or 2 × 449
  • 898 has no square factors that allow its square root to be simplified. √898 ≈ 29.9666

 

888 and Level 2

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888 consists of three 8’s, so it is divisible by 3.

Print the puzzles or type the solution on this excel file: 12 factors 886-896

888 is the hypotenuse of a Pythagorean triple:

  • 288-840-888 which is 24 times (12-35-37)

888 is the sum of eight consecutive prime numbers:

  • 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 = 888

888 is a palindrome in BASE 15 and a repdigit in BASE 36:

  • 3E3 BASE 15 (E is 14 in base 10), because 3(15²) + 14(15) + 3(1) = 888
  • OO BASE 36 (O is 24 in base 10), because 24(26) + 24(1) = 24(27) = 888

How many factors does 888 have?

  • 888 is a composite number.
  • Prime factorization: 888 = 2 × 2 × 2 × 3 × 37, which can be written 888 = 2³ × 3 × 37
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 888 has exactly 16 factors.
  • Factors of 888: 1, 2, 3, 4, 6, 8, 12, 24, 37, 74, 111, 148, 222, 296, 444, 888
  • Factor pairs: 888 = 1 × 888, 2 × 444, 3 × 296, 4 × 222, 6 × 148, 8 × 111, 12 × 74, or 24 × 37
  • Taking the factor pair with the largest square number factor, we get √888 = (√4)(√222) = 2√222 ≈ 29.7993.

877 and Level 2

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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.

 

867 and Level 2

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867 is composed of three consecutive numbers so 867 is divisible by 3. The middle number of those three numbers, 6, 7, 8 is 7 so 867 is NOT divisible by 9.

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

867 is the hypotenuse of two Pythagorean triples:

  • 483-720-867, which is 3 times (161-240-289)
  • 408-765-867 which is (8-15-17) times 51

867 is 300 in BASE 17 because 3(17²) = 867.

  • 867 is a composite number.
  • Prime factorization: 867 = 3 × 17 × 17, which can be written 867 = 3 × 17²
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 × 3  = 6. Therefore 867 has exactly 6 factors.
  • Factors of 867: 1, 3, 17, 51, 289, 867
  • Factor pairs: 867 = 1 × 867, 3 × 289, or 17 × 51
  • Taking the factor pair with the largest square number factor, we get √867 = (√289)(√3) = 17√3 ≈ 29.44486

855 A Bottle Full of Multiplication Facts

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If you’ve always wanted to know the multiplication facts better, there is hope for you to do that in this bottle! Just write the numbers from 1 to 10 in the top row and also in the first column in an order that makes those factors and the given clues act like a multiplication table.

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

855 is the hypotenuse of Pythagorean triple 513-684-855 which is (3, 4, 5) times 171.

From OEIS.org I learned that 855 can be expressed as sum of five consecutive squares (11² + 12² + 13² + 14² + 15² = 855) and the sum of two consecutive cubes (7³ + 8³ = 855). 855 is the smallest number that can make such a claim.

  • 855 is a composite number.
  • Prime factorization: 855 = 3 × 3 × 5 × 19, which can be written 855 = 3² × 5 × 19
  • 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 855 has exactly 12 factors.
  • Factors of 855: 1, 3, 5, 9, 15, 19, 45, 57, 95, 171, 285, 855
  • Factor pairs: 855 = 1 × 855, 3 × 285, 5 × 171, 9 × 95, 15 × 57, or 19 × 45,
  • Taking the factor pair with the largest square number factor, we get √855 = (√9)(√95) = 3√95 ≈ 29.240383

846 and Level 2

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

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

846 can be written as the sum of consecutive prime numbers two different ways. Together they use ALL the prime numbers from 13 to 127 exactly one time.

  • 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83 = 846; that’s eighteen consecutive primes.
  • 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 = 846; that’s eight consecutive primes.

OEIS.org informs us that 846² = 715,716. Not quite a Ruth Aaron pair, but still quite impressive.

  • 846 is a composite number.
  • Prime factorization: 846 = 2 × 3 × 3 × 47, which can be written 846 = 2 × 3² × 47
  • The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 846 has exactly 12 factors.
  • Factors of 846: 1, 2, 3, 6, 9, 18, 47, 94, 141, 282, 423, 846
  • Factor pairs: 846 = 1 × 846, 2 × 423, 3 × 282, 6 × 141, 9 × 94, or 18 × 47
  • Taking the factor pair with the largest square number factor, we get √846 = (√9)(√94) = 2√209 ≈ 29.086079

Finding Ways to Write 836 as the Sum of Consecutive Numbers

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OEIS.org informs us that 836² = 698,896, a palindrome.

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

836 can be written as the sum of 11 consecutive numbers and as the sum of 19 consecutive numbers because 11 and 19 are its odd factors (not including 1) that are less than 41. (Remember 861 is the 41st triangular number.) Notice 836’s factor pairs highlighted in red.

  • 71 + 72 + 73 + 74 + 75 + 76 + 77 + 78 + 79 + 80 + 81 = 836; that’s 11 consecutive numbers
  • 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 = 836; that’s 19 consecutive numbers

836 can be written as the sum of 8 consecutive numbers. Why? Because its factor that is the greatest power of 2 is 4, and because 1 is a factor of 836. Note that 2(4)(1) = 8.

  • 101 + 102 + 103 + 104 + 105 + 106 + 107 + 108 = 836

836 can’t be written as the sum of 2(4)(11) = 88 consecutive numbers or 2(4)(19) = 152 consecutive numbers because neither 88 or 152 is less than 41.

  • 836 is a composite number.
  • Prime factorization: 836 = 2 × 2 × 5 × 41, which can be written 836 = 2² × 11 × 19
  • 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 836 has exactly 12 factors.
  • Factors of 836: 1, 2, 4, 11, 19, 22, 38, 44, 76, 209, 418, 836
  • Factor pairs: 836 = 1 × 836, 2 × 418, 4 × 209, 11 × 76, 19 × 44, or 22 × 38
  • Taking the factor pair with the largest square number factor, we get √836 = (√4)(√209) = 2√209 ≈ 28.91366

830 I Can Divide These Polynomials By (x – 2) Without Even Looking at Them

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

(x – 2) is a factor of an infinite number of polynomials. I am listing only a small, but very special subset of them here. First look for the pattern that allows us to generate a polynomial from a given number in base 2. Then look for another pattern when the polynomial is divided by (x – 2).

Do you see the patterns? I do.

From the first pattern I know there is a similar special polynomial that ends with -830. AND I know from the second pattern what I will get if I divide THAT polynomial by (x – 2). Now get this: Even though I haven’t seen the polynomial yet, I know what the quotient will be! When the polynomial ending in -830 is divided by (x – 2), it will be. . . .

  • x⁸ + 3x⁷ + 6x⁶ + 12x⁵ + 25x⁴ + 51x³ + 103x² + 207x + 415

And guess what, I’m right! I found the quotient without showing any steps or even looking at what I was dividing.

How did I know what that quotient would be without writing down the problem and doing some division first? Well, not only is this polynomial special, but the quotient is special, too!

All I needed to know was that the last term was -830. I then divided 830 repeatedly by 2. Any time my quotient was an odd number, I subtracted one from it before I divided it again by 2. I repeated the process until I reached zero. That is how I got all my coefficients. Even though I could do this problem without showing any work, I made a gif so you and anyone else can quickly see how I did it, but you’ll have to look sideways at it to see it. Showing steps is ALWAYS a good thing.
Find 830 in BASE 2

make science GIFs like this at MakeaGif
As a side benefit, this is another way to find out what 830 is in BASE 2.

Here’s a little more about the number 830:

  • 830 is a composite number.
  • Prime factorization: 830 = 2 × 5 × 83
  • 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 830 has exactly 8 factors.
  • Factors of 830: 1, 2, 5, 10, 83, 166, 415, 830
  • Factor pairs: 830 = 1 × 830, 2 × 415, 5 × 166, or 10 × 83
  • 830 has no square factors that allow its square root to be simplified. √830 ≈ 28.80972

830 is the sum of four consecutive prime numbers:

  • 197 + 199 + 211 + 223 = 830

Because 5 is one of its factors, 830 is the hypotenuse of a Pythagorean triple:

  • 498-664-830; that’s 166 times (3-4-5)