894 and Level 5

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894 is the hypotenuse of a Pythagorean triple: 306-840-894

894 is also palindrome 626 in BASE 12 because 6(144) + 2(12) + 6(1) = 894

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

  • 894 is a composite number.
  • Prime factorization: 894 = 2 × 3 × 149
  • 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 894 has exactly 8 factors.
  • Factors of 894: 1, 2, 3, 6, 149, 298, 447, 894
  • Factor pairs: 894 = 1 × 894, 2 × 447, 3 × 298, or 6 × 149
  • 894 has no square factors that allow its square root to be simplified. √894 ≈ 29.89983

A little surprise is waiting when you square 893.

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What is special about the number 893? It makes a pretty cool square with two 4’s, 7’s, and 9’s in it. Thank you OEIS.org for that fun fact.

  • 893 is a composite number.
  • Prime factorization: 893 = 19 × 47
  • 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 893 has exactly 4 factors.
  • Factors of 893: 1, 19, 47, 893
  • Factor pairs: 893 = 1 × 893 or 19 × 47
  • 893 has no square factors that allow its square root to be simplified. √893 ≈ 29.8831

892 Tribute to The Mysteries of Harris Burdick

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When I finished making today’s puzzle, I remembered a particular picture from The Mysteries of Harris Burdick. Can you guess which picture that would be?

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

This popular children’s book contains only pictures with short captions. Children use their imaginations to write short stories for the curious pictures and captions. The book is available in municipal libraries everywhere and on amazon.com.

892 looks interesting in a couple of different bases:

  • It is 4044 in BASE 6, because 4(6³) + 4(6) + 4(1) = 4(216 + 6 + 1) = 4(223) = 892
  • It is 161 in BASE 27, because 1(27²) + 6(27) + 1(1) = 892

892 is also the sum of consecutive prime numbers: 443 + 449 = 892

  • 892 is a composite number.
  • Prime factorization: 892 = 2 × 2 × 223, which can be written 892 = 2² × 223
  • 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 892 has exactly 6 factors.
  • Factors of 892: 1, 2, 4, 223, 446, 892
  • Factor pairs: 892 = 1 × 892, 2 × 446, or 4 × 223
  • Taking the factor pair with the largest square number factor, we get √892 = (√4)(√223) = 2√223 ≈ 29.866369

891 Mystery Level Puzzle

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Sometimes revealing the puzzle level reveals more than is needed. I think I will periodically publish a Mystery Level puzzle. Can you solve this one?

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

8 + 9 + 1 = 18, so 891 is divisible by 3 and by 9.

8 – 9 + 1 = 0, so 891 is divisible by 11.

891 looks interesting in a few different bases:

  • 1(2⁹) + 1(2⁸) + 0(2⁷) + 1(2⁶) + 1(2⁵) + 1(2⁴) + 1(2³) + 0(2²) + 1(2¹) + 1(2⁰) =891, so it’s palindrome 1101111011 in BASE 2.
  • RR in BASE 32 (R is 27 in base 10), because 27(32) + 27(1) = 27(33) = 891
  • R0 in BASE 33, because 27(33) = 891

891 is also the sum of five consecutive prime numbers: 167 + 173 + 179 + 181 + 191 = 891

  • 891 is a composite number.
  • Prime factorization: 891 = 3 × 3 × 3 × 3 × 11, which can be written 891 = 3⁴ × 11
  • 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 891 has exactly 10 factors.
  • Factors of 891: 1, 3, 9, 11, 27, 33, 81, 99, 297, 891
  • Factor pairs: 891 = 1 × 891, 3 × 297, 9 × 99, 11 × 81, or 27 × 33
  • Taking the factor pair with the largest square number factor, we get √891 = (√81)(√11) = 9√11 ≈ 29.849623

891 is in this cool pattern:

 

890 and Level 4

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890 is the sum of four consecutive prime numbers:

  • 211 + 223 + 227 + 229 = 890

890 is also the hypotenuse of four Pythagorean triples:

  • 168-874-890, which is 2 times (84-437-445)
  • 390-800-890, which is 10 times (39-80-89)
  • 406-792-890, which is 2 times (203-396-445)
  • 534-712-890, which is (3-4-5) times 178

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

  • 890 is a composite number.
  • Prime factorization: 890 = 2 × 5 × 89
  • 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 890 has exactly 8 factors.
  • Factors of 890: 1, 2, 5, 10, 89, 178, 445, 890
  • Factor pairs: 890 = 1 × 890, 2 × 445, 5 × 178, or 10 × 89
  • 890 has no square factors that allow its square root to be simplified. √890 ≈ 29.83286778.

889 and Level 3

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889 is a palindrome in bases 13 and 24:

It is 535 in BASE 13 because 5(13²) + 3(13) + 5(1) = 889.

It is 1D1 in BASE 24 (D is 13 base 10) because 1(24²) + 13(24) + 1(1) = 889.

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

  • 889 is a composite number.
  • Prime factorization: 889 = 7 × 127
  • 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 889 has exactly 4 factors.
  • Factors of 889: 1, 7, 127, 889
  • Factor pairs: 889 = 1 × 889 or 7 × 127
  • 889 has no square factors that allow its square root to be simplified. √889 ≈ 29.816103

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.

887 is the Last Prime Number for a While!

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The last prime number before 887 was 883.

The next prime number won’t be until 907. Wow!

907 – 887 = 20. That’s the largest gap between prime numbers so far!

887 is also a palindrome in a few other bases:

  • 31313 BASE 4, because 3(4⁴) + 1(4³) + 3(4²) + 1(4) + 3(1) = 887
  • 737 BASE 11, because 7(11²) + 3(11) + 7(1) = 887
  • 313 BASE 17, because 3(17²) + 1(17) + 3(1) = 887

Those patterns of 3’s and 1’s for two different bases surprised me!

  • 887 is a prime number.
  • Prime factorization: 887 is prime.
  • The exponent of prime number 887 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 887 has exactly 2 factors.
  • Factors of 887: 1, 887
  • Factor pairs: 887 = 1 × 887
  • 887 has no square factors that allow its square root to be simplified. √887 ≈ 29.782545

How do we know that 887 is a prime number? If 887 were not a prime number, then it would be divisible by at least one prime number less than or equal to √887 ≈ 29.8. Since 887 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 887 is a prime number.

 

886 and Level 1

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886 is the sum of the sixteen prime numbers from 23 to 89.

886 is a palindrome in a couple of other bases:

  • 12021 in BASE 5, because 1(5⁴) + 2(5³) + 0(5²) + 2(5) + 1(1) = 886
  • 474 in BASE 14, because 4(14²) + 7(14) + 4(1) = 886

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

  • 886 is a composite number.
  • Prime factorization: 886 = 2 × 443
  • 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 886 has exactly 4 factors.
  • Factors of 886: 1, 2, 443, 886
  • Factor pairs: 886 = 1 × 886 or 2 × 443
  • 886 has no square factors that allow its square root to be simplified. √886 ≈ 29.76575

 

885 and Level 6

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885 obviously is divisible by 5. Since it is exactly 3 numbers away from 888, it also can be evenly divided by 3 so adding up all its digits was not necessary to test for divisibility.

885 is the hypotenuse of a Pythagorean triple:

  • 531-708-885 which is (3-4-5) times 177

885 is palindrome 181 in BASE 26 because 1(26²) + 8(26) + 1(1) = 885

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

  • 885 is a composite number.
  • Prime factorization: 885 = 3 × 5 × 59
  • 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 885 has exactly 8 factors.
  • Factors of 885: 1, 3, 5, 15, 59, 177, 295, 885
  • Factor pairs: 885 = 1 × 885, 3 × 295, 5 × 177, or 15 × 59
  • 885 has no square factors that allow its square root to be simplified. √885 ≈ 29.74894956