Factors

1222 and Level 2

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Can you write the numbers from 1 to 10 in both the first column and the top row of the table below so that the given clues are the multiplication products of the factors you wrote? There is only one solution, but I am sure that you can find it.

Print the puzzles or type the solution in this excel file: 10-factors-1221-1231

Now I’ll write a little bit about the number 1222:

  • 1222 is a composite number.
  • Prime factorization: 1222 = 2 × 13 × 47
  • 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 1222 has exactly 8 factors.
  • Factors of 1222: 1, 2, 13, 26, 47, 94, 611, 1222
  • Factor pairs: 1222 = 1 × 1222, 2 × 611, 13 × 94, or 26 × 47
  • 1222 has no square factors that allow its square root to be simplified. √1222 ≈ 34.95712

1222 is the hypotenuse of a Pythagorean triple:
470-1128-1222 which is (5-12-13) times 94

1221 and Level 1

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This puzzle is like a multiplication table with its factors in a different order. Can you figure out where the factors from 1 to 10 go in both the first column and the top row? Afterward, can you correctly fill in every cell of this mixed-up multiplication table?

Print the puzzles or type the solution in this excel file: 10-factors-1221-1231

Let me share some facts about the number 1221:

  • 1221 is a composite number.
  • Prime factorization: 1221 = 3 × 11 × 37
  • 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 1221 has exactly 8 factors.
  • Factors of 1221: 1, 3, 11, 33, 37, 111, 407, 1221
  • Factor pairs: 1221 = 1 × 1221, 3 × 407, 11 × 111, or 33 × 37
  • 1221 has no square factors that allow its square root to be simplified. √1221 ≈ 34.94281

1 × 11 × 111 = 1221

1221 is the sum of five consecutive prime numbers:
233 + 239 + 241 + 251 + 257 = 1221

1221 is the hypotenuse of a Pythagorean triple:
396-1155-1221 which is 33 times (12-35-37)

Not only is 1221 a palindrome in base 10 but look at it in these other bases:
It’s 14341 in BASE 5,
5353 in BASE 6,
272 in BASE 23, and
it’s XX in BASE 36 because 33(36) + 33(1) = 33(37) = 1221

1220 Challenge Puzzle

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The last challenge puzzle was particularly difficult. This one won’t be nearly as bad. Try it and see if you can figure it out!

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Here are a few facts about the number 1220:

  • 1220 is a composite number.
  • Prime factorization: 1220 = 2 × 2 × 5 × 61, which can be written 1220 = 2² × 5 × 61
  • 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 1220 has exactly 12 factors.
  • Factors of 1220: 1, 2, 4, 5, 10, 20, 61, 122, 244, 305, 610, 1220
  • Factor pairs: 1220 = 1 × 1220, 2 × 610, 4 × 305, 5 × 244, 10 × 122, or 20 × 61
  • Taking the factor pair with the largest square number factor, we get √1220 = (√4)(√305) = 2√305 ≈ 34.9285

1220 is the sum of consecutive prime numbers: 607 + 613 = 1220

1220 is the sum of two squares two different ways:
32² + 14² = 1220
34² + 8² = 1220

1220 is the hypotenuse of four Pythagorean triples:
220-1200-1220 which is 20 times (11-60-61)
544-1092-1220 calculated from 2(34)( 8), 34² – 8², 34² + 8²
and is also 4 times (136-273-305)
828-896-1220 calculated from 32² – 14², 2(32)(14), 32² + 14²
and is also 4 times (207-224-305)
732-976-1220 which is (3-4-5) times 244

1219 is a Centered Triangular Number

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If you look at an ordered list of centered triangular numbers, 1219 will be the 29th number on the list.

Study this graphic to see why:

Here’s more about the number 1219:

  • 1219 is a composite number.
  • Prime factorization: 1219 = 23 × 53
  • 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 1219 has exactly 4 factors.
  • Factors of 1219: 1, 23, 53, 1219
  • Factor pairs: 1219 = 1 × 1219 or 23 × 53
  • 1219 has no square factors that allow its square root to be simplified. √1219 ≈ 34.91418

1219 is the sum of the thirteen prime numbers from 67 to 127.

1219 is also the hypotenuse of a Pythagorean triple:
644-1035-1219 which is 23 times (28-45-53)

 

1218 Mystery Level

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Working on this Mystery level puzzle may seem easy at first, but will it remain easy until its unique solution is found? Only those who solve it will know the answer to that question!

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Here are some facts about the number 1218:

  • 1218 is a composite number.
  • Prime factorization: 1218 = 2 × 3 × 7 × 29
  • 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 1218 has exactly 16 factors.
  • Factors of 1218: 1, 2, 3, 6, 7, 14, 21, 29, 42, 58, 87, 174, 203, 406, 609,
  • 1218
  • Factor pairs: 1218 = 1 × 1218, 2 × 609, 3 × 406, 6 × 203, 7 × 174, 14 × 87, 21 × 58, or 29 × 42
  • 1218 has no square factors that allow its square root to be simplified. √1218 ≈ 34.89986

1218 is the hypotenuse of a Pythagorean triple:
840-882-1218 which is (20-21-29) times 42

 

1217 Squeegee Mystery

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The murder weapon for today’s mystery appears to be a squeegee! Study all the clues in the puzzle and see if you can find all the needed factors to solve this mystery. As in every whodunnit, there is only one true solution.

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Now here are a few facts about the number 1217:

  • 1217 is a prime number.
  • Prime factorization: 1217 is prime.
  • The exponent of prime number 1217 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1217 has exactly 2 factors.
  • Factors of 1217: 1, 1217
  • Factor pairs: 1217 = 1 × 1217
  • 1217 has no square factors that allow its square root to be simplified. √1217 ≈ 34.88553

How do we know that 1217 is a prime number? If 1217 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1217 ≈ 34.9. Since 1217 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1217 is a prime number.

31² + 16² = 1217

1217 is the hypotenuse of a Pythagorean triple:
705-992-1217 calculated from 31² – 16², 2(31)(16), 31² + 16²

Here’s another way we know that 1217 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 16² = 1217 with 31 and 16 having no common prime factors, 1217 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1217 ≈ 34.9. Since 1217 is not divisible by 5, 13, 17, or 29, we know that 1217 is a prime number.

1216 and Level 6

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Is 6 or 9 the needed common factor for 54 and 36?
The other one will be the common factor for 72 and 18.

Will 5 or 10 be the common factor of 30 and 50?
Will 2 or 4 be the common factor of 16 and 4?

Don’t guess the answers to those questions! The other clues in the puzzle will help you find the answers. Will you be stumped or will you triumph?

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Now I’ll share some information about the number 1216:

  • 1216 is a composite number.
  • Prime factorization: 1216 = 2 × 2 × 2 × 2 × 2 × 2 × 19, which can be written 1216 = 2⁶ × 19
  • The exponents in the prime factorization are 6, and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1) = 7 × 2 = 14. Therefore 1216 has exactly 14 factors.
  • Factors of 1216: 1, 2, 4, 8, 16, 19, 32, 38, 64, 76, 152, 304, 608, 1216
  • Factor pairs: 1216 = 1 × 1216, 2 × 608, 4 × 304, 8 × 152, 16 × 76, 19 × 64, or 32 × 38
  • Taking the factor pair with the largest square number factor, we get √1216 = (√64)(√19) = 8√19 ≈ 34.87119

1216 is the sum of the ten prime numbers between 100 and 150:
101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 = 1216

1216 is the 19th nonagonal number because
19(7·19 – 5)/2 = 1216, in other words because 19 × 64 = 1216

1215 and Level 5

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Level 5 puzzles are a little trickier than level 4 puzzles, but you can still use logic to find the unique solution to today’s puzzle. Stick with it!

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Here is some information about the number 1215:

  • 1215 is a composite number.
  • Prime factorization: 1215 = 3 × 3 × 3 × 3 × 3 × 5, which can be written 1215 = 3⁵ × 5
  • The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 1215 has exactly 12 factors.
  • Factors of 1215: 1, 3, 5, 9, 15, 27, 45, 81, 135, 243, 405, 1215
  • Factor pairs: 1215 = 1 × 1215, 3 × 405, 5 × 243, 9 × 135, 15 × 81, or 27 × 45
  • Taking the factor pair with the largest square number factor, we get √1215 = (√81)(√15) = 9√15 ≈ 34.85685

1215 is the hypotenuse of a Pythagorean triple:
729-972-1215 which is (3-4-5) times 243

 

1214 and Level 4

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Even though I don’t tell you the order to consider the clues in today’s puzzle, you can still solve it. Just use logic and your knowledge of a standard 12 × 12 multiplication table to find its unique solution. Good luck!

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Now I’ll share a few facts about the number 1214:

  • 1214 is a composite number.
  • Prime factorization: 1214 = 2 × 607
  • 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 1214 has exactly 4 factors.
  • Factors of 1214: 1, 2, 607, 1214
  • Factor pairs: 1214 = 1 × 1214 or 2 × 607
  • 1214 has no square factors that allow its square root to be simplified. √1214 ≈ 34.8425

1214 is the sum of six consecutive prime numbers:
191 + 193 + 197 + 199 + 211 + 223 = 1214

1213 and Level 3

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This puzzle would be a lot tougher to solve if I didn’t put the clues in the order that I did. Just start at the top of the puzzle and work your way down the puzzle clue by clue until you get to the bottom of the puzzle and have the entire thing solved.

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Now I’ll share a few facts about the number 1213:

  • 1213 is a prime number.
  • Prime factorization: 1213 is prime.
  • The exponent of prime number 1213 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1213 has exactly 2 factors.
  • Factors of 1213: 1, 1213
  • Factor pairs: 1213 = 1 × 1213
  • 1213 has no square factors that allow its square root to be simplified. √1213 ≈ 34.82815

How do we know that 1213 is a prime number? If 1213 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1213 ≈ 34.8. Since 1213 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1213 is a prime number.

1213 is the sum of these nine consecutive prime numbers:
109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 = 1213

27² + 22² = 1213
1213 is the hypotenuse of a Pythagorean triple:
245-1188-1213 calculated from 27² – 22², 2(27)(22), 27² + 22²

Here’s another way we know that 1213 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 27² + 22² = 1213 with 27 and 22 having no common prime factors, 1213 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1213 ≈ 34.8. Since 1213 is not divisible by 5, 13, 17, or 29, we know that 1213 is a prime number.