Puzzles

865 and Level 1

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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

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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.

858 and Level 4

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There are sixteen numbers less than 1000 that have four different prime factors. 858 is one of them, and it is the ONLY one that is also a palindrome. Thank you, OEIS.org for alerting us to that fact. No smaller palindrome has four different prime factors!

The sixteen products on that chart each have exactly sixteen factors!

Here’s a Find the Factors 1-10 puzzle for you to solve:

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

Here’s a little more about the number 858:

858 is the hypotenuse of a Pythagorean triple: 330-792-858

  • 858 is a composite number.
  • Prime factorization: 858 = 2 × 3 × 11 × 13
  • The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 858 has exactly 16 factors.
  • Factors of 858: 1, 2, 3, 6, 11, 13, 22, 26, 33, 39, 66, 78, 143, 286, 429, 858
  • Factor pairs: 858 = 1 × 858, 2 × 429, 3 × 286, 6 × 143, 11 × 78, 13 × 66, 22 × 39, or 26 × 33
  • 858 has no square factors that allow its square root to be simplified. √858 ≈ 29.291637

 

 

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, …

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

853 You Can Do This Puzzle!

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This is a level 1 puzzle that is easier than even most other level 1 puzzles. You can do this puzzle! If mathematics makes you uncomfortable, you can still do this puzzle! Even if math class is your worse nightmare, you can complete this puzzle, and gain a little confidence. Go ahead, give it a try! Figure out where each number from one to ten goes in the top row and also in the first column so that the puzzle turns into a mixed-up multiplication table. It’s easier and far less time consuming than Sudoku. You CAN do this puzzle! Then, after you find all the factors, and are feeling really good about yourself, IF you want, you can fill in all the other cells of this mixed up multiplication table.

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

853 is a prime number that leaves a remainder of 1 when divided by 4, so 853 is the hypotenuse of a Pythagorean triple: 205-828-853.

23² + 18² = 853 so 205-828-853 can be calculated from 23² – 18², 2(23)(18), 23² + 18².

  • 853 is a prime number.
  • Prime factorization: 853 is prime.
  • The exponent of prime number 853 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 853 has exactly 2 factors.
  • Factors of 853: 1, 853
  • Factor pairs: 853 = 1 × 853
  • 853 has no square factors that allow its square root to be simplified. √853 ≈ 29.20616

How do we know that 853 is a prime number? If 853 were not a prime number, then it would be divisible by at least one prime number less than or equal to √853 ≈ 29.2. Since 853 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 853 is a prime number.

Here’s another way we know that 853 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 23² + 18² = 853 with 23 and 18 having no common prime factors, 853 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √853 ≈ 29.2. Since 853 is not divisible by 5, 13, 17, or 29, we know that 853 is a prime number.

 

852 and Level 6

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

I knew that 852 was divisible by 3 as soon as I typed it in a straight line on the number pad. Any 3 digit number that lies on a straight line on a number pad or a phone dial pad is divisible by 3. And in case you’ve ever wondered why the numbers on a number pad or calculator and the numbers on a phone dial pad are reversed, ABC News has the answer.

852 is 705 in BASE 11, and it is 507 in BASE 13.

852 is palindrome 1E1 in BASE 23 (E is 14 base 10) because 1(23²) +14(23¹) + 1(23º) = 852.

852 is the sum of consecutive prime numbers 421 and 431.

852 is also the 24th pentagonal number because (3⋅24² – 24)/2 = 852

  • 852 is a composite number.
  • Prime factorization: 852 = 2 × 2 × 3 × 71, which can be written 852 = 2² × 3 × 71
  • 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 852 has exactly 12 factors.
  • Factors of 852: 1, 2, 3, 4, 6, 12, 71, 142, 213, 284, 426, 852
  • Factor pairs: 852 = 1 × 852, 2 × 426, 3 × 284, 4 × 213, 6 × 142, or 12 × 71,
  • Taking the factor pair with the largest square number factor, we get √852 = (√4)(√213) = 2√213 ≈ 29.189039

851 Give This Apple to Your Teacher This Year

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This puzzle looks a little like an apple. It’s a level 5 puzzle so it won’t be that easy. If you can solve the puzzle, give it to your teacher!


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

851 is the hypotenuse of a Pythagorean triple:

  • 276-805-851 which is 23 times (12-35-37)

851 is a palindrome in three other bases:

  • 353 BASE 16, because 3(16²) + 5(16¹) + 3(16º) = 851
  • 191 BASE 25, because 1(25²) + 9(25¹) + 1(25º) = 851
  • NN BASE 36 (N is 23 base 10) because 23(36¹) + 23(36º) = 23(37) = 851

Here is 851 factoring information:

  • 851 is a composite number.
  • Prime factorization: 851 = 23 × 37
  • 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 851 has exactly 4 factors.
  • Factors of 851: 1, 23, 37, 851
  • Factor pairs: 851 = 1 × 851 or 23 × 37
  • 851 has no square factors that allow its square root to be simplified. √851 ≈ 171904

851 is in this cool pattern:

 

849 and Level 4

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

8 + 4 + 9 = 21, a number divisible by 3 so odd number 849 can be evenly divided by 3, AND 849 can be written as the sum of three consecutive numbers and six consecutive numbers.

  • 282 + 283 + 284 = 849.
  • 139 + 140 + 141 + 142 + 143 + 144 = 849

849 can also be written as the sum of three odd numbers: 281 + 283 + 285 = 849.

849 can be written as the difference of two squares two different ways:

  • 143² – 140² = 849
  • 425² – 424² = 849

True, similar things can be written about every other number that is divisible by 3, but they are still fun facts about the number 849.

  • 849 is a composite number.
  • Prime factorization: 849 = 3 × 283
  • 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 849 has exactly 4 factors.
  • Factors of 849: 1, 3, 283, 849
  • Factor pairs: 849 = 1 × 849 or 3 × 283
  • 849 has no square factors that allow its square root to be simplified. √849 ≈ 29.1376