Puzzle

877 and Level 2

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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

/ ivasallay / Leave a comment

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.

873 and Level 5

/ ivasallay / 2 Comments

8 + 7 + 3 = 18; 1 + 8 = 9, so 873 can be evenly divided by 9.

27² + 12² = 873 so 873 is the hypotenuse of a Pythagorean triple:

  • 585-648-873 which is 9 times (65-72-97), and can be calculated from 27² – 12², 2(27)(12), 27² + 12²

OEIS.org reminds us that 1! + 2! + 3! + 4! + 5! + 6! = 873.

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

  • 873 is a composite number.
  • Prime factorization: 873 = 3 × 3 × 97, which can be written 873 = 3² × 97
  • 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 873 has exactly 6 factors.
  • Factors of 873: 1, 3, 9, 97, 291, 873
  • Factor pairs: 873 = 1 × 873, 3 × 291, or 9 × 97
  • Taking the factor pair with the largest square number factor, we get √873 = (√9)(√97) = 3√97 ≈ 29.546573

 

872 and Level 4

/ ivasallay / Leave a comment

26² + 14² = 872 so 872 is the hypotenuse of a Pythagorean triple:

480-728-872 which can be calculated from 26² – 14², 2(26)(14), 26² + 14².

872 is the sum of consecutive prime numbers 433 and 439.

872 is also the sum of the 21 prime numbers from 3 to 79.

872! + 1 is a number much too big for any calculator I own, but OEIS.org informs us that it is a prime number.

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

 

 

  • 872 is a composite number.
  • Prime factorization: 872 = 2 × 2 × 2 × 109, which can be written 872 = 2³ × 109
  • 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 872 has exactly 8 factors.
  • Factors of 872: 1, 2, 4, 8, 109, 218, 436, 872
  • Factor pairs: 872 = 1 × 872, 2 × 436, 4 × 218, or 8 × 109
  • Taking the factor pair with the largest square number factor, we get √872 = (√4)(√218) = 2√218 ≈ 29.529646

869 and Level 3

/ ivasallay / Leave a comment

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

867 and Level 2

/ ivasallay / Leave a comment

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

865 and Level 1

/ ivasallay / 2 Comments

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

865 is the sum of two squares two different ways:

  • 28² + 9² = 865
  • 24² + 17² = 865

865 is the hypotenuse of four Pythagorean triples, two of which are primitives:

  • 260-825-865, which is 5 times (52-165-173)
  • 287-816-865, which is 24² – 17², 2(24)(17), 24² + 17²
  • 504-703-865 which is 2(28)(9), 28² – 9², 28² – 9²
  • 519-692-865, which is (3-4-5) times 173

You could see 865’s factors in two of those Pythagorean triples, and here they are again:

  • 865 is a composite number.
  • Prime factorization: 865 = 5 × 173
  • 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 865 has exactly 4 factors.
  • Factors of 865: 1, 5, 173, 865
  • Factor pairs: 865 = 1 × 865 or 5 × 173
  • 865 has no square factors that allow its square root to be simplified. √865 ≈ 29.41088

860 and Level 6

/ ivasallay / Leave a comment

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

860 is the hypotenuse of a Pythagorean triple: 516-688-860, which is (3-4-5) times 172.

860 can be written as the sum of four consecutive prime numbers: 199 + 211 + 223 + 227 = 860

  • 860 is a composite number.
  • Prime factorization: 860 = 2 × 2 × 5 × 43, which can be written 860 = 2² × 5 × 43
  • 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 860 has exactly 12 factors.
  • Factors of 860: 1, 2, 4, 5, 10, 20, 43, 86, 172, 215, 430, 860
  • Factor pairs: 860 = 1 × 860, 2 × 430, 4 × 215, 5 × 172, 10 × 86, or 20 × 43,
  • Taking the factor pair with the largest square number factor, we get √860 = (√4)(√215) = 2√215 ≈ 29.3257566

What Kind of Prime Is 859?

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A prime number is a positive number that has exactly two factors, one and itself. (One has only one factor, so it is not a prime number.)

  • 859 is the 149th prime number.

A twin prime is a set of two prime numbers in which the second prime number is two more that the first prime number.

  • 859 is the second prime number in the 34th twin prime: (857, 859).

A prime triplet is a set of three consecutive prime numbers in which the last number is six more than the first number. Prime triplets always contain a set of twin primes.

  • 859 is in the 27th and 28th prime triplets: (853, 857, 859) and (857, 859, 863).

A prime quadruplet is a set of four consecutive prime numbers in which the last number is eight more than the first number. Prime quadruplets always contain TWO sets of overlapping prime triplets.

  • Even though prime numbers (853, 857, 859, 863) contain two sets of overlapping prime triplets, they do NOT form a prime quadruplet because the last number is ten more than the first number. Other than (5, 7, 11, 13), all prime quadruplets are prime decades whose last digits are 1, 3, 7, and 9, in THAT order.

There are other prime constellations like prime quintuplets and prime sextuplets, but each of those has to contain a prime quadruplet in it, so 859 isn’t in any of those.

859÷4 = 214 R3. Since that wasn’t R1, we know that 859 is NOT the hypotenuse of ANY Pythagorean triples.

Now you know what kind of prime 859 is.

Here’s today’s puzzle:

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

  • 859 is a prime number.
  • Prime factorization: 859 is prime.
  • The exponent of prime number 859 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 859 has exactly 2 factors.
  • Factors of 859: 1, 859
  • Factor pairs: 859 = 1 × 859
  • 859 has no square factors that allow its square root to be simplified. √859 ≈ 29.3087

How do we know that 859 is a prime number? If 859 were not a prime number, then it would be divisible by at least one prime number less than or equal to √859 ≈ 29.3. Since 859 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 859 is a prime number.