Level 5 Puzzle

938 and Level 5

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Can you figure out where the factors 1 – 10 go in the first column and top row so that this level 5 puzzle will function as a multiplication table?

Print the puzzles or type the solution on this excel file: 10-factors-932-941

Now I’ll share a few facts about the number 938.

938 is a palindrome in two consecutive bases:
It’s 343 in BASE 17 because 3(17²) + 4(17¹) + 3(17º) = 938
It’s 2G2 in BASE 18 (G is 16 base 10), because 2(18²) + 16(18¹) + 2(18º) = 938

  • 938 is a composite number.
  • Prime factorization: 938 = 2 × 7 × 67
  • 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 938 has exactly 8 factors.
  • Factors of 938: 1, 2, 7, 14, 67, 134, 469, 938
  • Factor pairs: 938 = 1 × 938, 2 × 469, 7 × 134, or 14 × 67
  • 938 has no square factors that allow its square root to be simplified. √938 ≈ 30.62678566

929 Little Green Monster

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Here’s a little green monster just in time for Halloween. It’s a level 5 so it might be a little scary. Just don’t write any of the factors in the first column or top row unless you know for sure that factor belongs where you are putting it. Use logic and not guessing, and you’ll handle this little green monster just fine.

Print the puzzles or type the solution on this excel file: 12 factors 923-931

929 is the sum of nine consecutive prime numbers:
83  + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 = 929

23² + 20² = 929, so 929 is the hypotenuse of a Pythagorean triple:
129-920-929 which is 23² – 20², 2(23)(20), 23² + 20²

Obviously 929 is a palindrome in base 10.

It is also palindrome 131 in BASE 29 because 1(29²) + 3(29) + 1(1) = 929.

  • 929 is a prime number.
  • Prime factorization: 929 is prime.
  • The exponent of prime number 929 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 929 has exactly 2 factors.
  • Factors of 929: 1, 929
  • Factor pairs: 929 = 1 × 929
  • 929 has no square factors that allow its square root to be simplified. √929 ≈ 30.4795013

How do we know that 929 is a prime number? If 929 were not a prime number, then it would be divisible by at least one prime number less than or equal to √929 ≈ 30.4. Since 929 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 929 is a prime number.

921 Is This Bug Cute or Creepy?

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Some bugs make creepy Halloween decorations. Other bugs, like ladybugs, might make a very cute costume.

Today’s puzzle looks like a bug, but there is no reason to run and hide from this one. Yes, it’s a level 5, so some parts of it may be tricky.

This is what you need to do to solve it: stay calm; don’t guess and check. Figure out where to put each number from 1 to 10 in both the top row and the first column so that the clues make the puzzle work like a multiplication table. Don’t write a number down unless you are absolutely sure it belongs where you’re putting it. Use logic, step by step, and this puzzle will be a treat.

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

When you put on a Halloween costume, you may look completely different.

When a number is written in a different base, it may look completely different. For example,
921 looks like repdigit 333 in BASE 17 because 3(17²) + 3(17¹) + 3(17º) = 3(289 + 17 + 1) = 3(307) = 921
(307 is 111 in BASE 17)

921 looks like palindrome 1H1 in BASE 23 (H is 17 base 10). As you might suspect, 1(23²) + 17(23¹) + 1(23º) = 529 + 391 + 1 = 921

When it’s not written in a different base, 921 looks pretty familiar. You can tell quite quickly that it is divisible by 3:

  • 921 is a composite number.
  • Prime factorization: 921 = 3 × 307
  • 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 921 has exactly 4 factors.
  • Factors of 921: 1, 3, 307, 921
  • Factor pairs: 921 = 1 × 921 or 3 × 307
  • 921 has no square factors that allow its square root to be simplified. √921 ≈ 30.34798181

910 and Level 5

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910 is the hypotenuse of four Pythagorean triples:

  • 224-882-910, which is 14 times (16-63-65).
  • 350-840-910, which is (5-12-13) times 70.
  • 462-784-910, which is 14 times (33-56-65)
  • 546-728-910, which is (3-4-5) times 182.

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

910 is 190 in BASE 26, and 910 looks interesting in some other bases, too:

  • 4114 in BASE 6, because 4(6³) + 1(6²) + 1(6¹) + 4(6º) = 910
  • 1221 in BASE 9, because 1(9³) + 2(9²) + 2(9¹) + 1(9º) = 910
  • QQ in BASE 34 (Q is 26 base 10), because 26(34¹) + 26(34º) = 26(34 + 1) = 26(35) = 910
  • 26 0 BASE 35, because 26(35) + 0(1) = 26(35) = 910

What are the factors of 910?

  • 910 is a composite number.
  • Prime factorization: 910 = 2 × 5 × 7 × 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 910 has exactly 16 factors.
  • Factors of 910: 1, 2, 5, 7, 10, 13, 14, 26, 35, 65, 70, 91, 130, 182, 455, 910
  • Factor pairs: 910 = 1 × 910, 2 × 455, 5 × 182, 7 × 130, 10 × 91, 13 × 70, 14 × 65, or 26 × 35
  • 910 has no square factors that allow its square root to be simplified. √910 ≈ 30.166206

 

902 and Level 5

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902 is the hypotenuse of a Pythagorean triple:

198-880-902 which is 22 times (9-40-41)

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

  • 902 is a composite number.
  • Prime factorization: 902 = 2 × 11 × 41
  • 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 902 has exactly 8 factors.
  • Factors of 902: 1, 2, 11, 22, 41, 82, 451, 902
  • Factor pairs: 902 = 1 × 902, 2 × 451, 11 × 82, or 22 × 41
  • 902 has no square factors that allow its square root to be simplified. √902 ≈ 30.0333.

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

883 and Level 5

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833 is the sum of the eleven prime numbers from 59 to 103.

833 is also the sum of these three consecutive prime numbers: 283 + 293 + 307 = 833.

883 is palindrome 373 in BASE 16 because 3(16²) + 7(16) + 3(1) = 883.

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

  • 883 is a prime number.
  • Prime factorization: 883 is prime.
  • The exponent of prime number 883 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 883 has exactly 2 factors.
  • Factors of 883: 1, 883
  • Factor pairs: 883 = 1 × 883
  • 883 has no square factors that allow its square root to be simplified. √883 ≈ 29.7153159

How do we know that 883 is a prime number? If 883 were not a prime number, then it would be divisible by at least one prime number less than or equal to √883 ≈ 29.7. Since 883 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 883 is a prime number.

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

 

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.

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: