Prime factorization

1070 A Complicated Logic Mystery

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This mystery level puzzle requires complicated logic just to get started. If you get stuck, ask yourself these questions:
What two clues must use both 5’s?
What two clues must, therefore, use both 6’s?
What do you know about clue 36 because it shares a common factor with 24?
What two clues must use both 3’s?
What do you now know about the common factor of clues 8 and 24?
What do you also now know about the common factor of clues 36 and 24?

Print the puzzles or type the solution in this excel file: 12 factors 1063-1072

Here is a little information about the number 1070:

  • 1070 is a composite number.
  • Prime factorization: 1070 = 2 × 5 × 107
  • 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 1070 has exactly 8 factors.
  • Factors of 1070: 1, 2, 5, 10, 107, 214, 535, 1070
  • Factor pairs: 1070 = 1 × 1070, 2 × 535, 5 × 214, or 10 × 107
  • 1070 has no square factors that allow its square root to be simplified. √1070 ≈ 32.71085

1070 is the hypotenuse of a Pythagorean triple:
642-856-1070 which is (3-4-5) times 214

 

1069 Mystery Level Puzzle

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It isn’t a mystery where the 7’s and the 11’s go in this puzzle, but then what are you going to do? That’s the mystery. Can you write each number from 1 to 12 in both the first column and the top row so that those numbers are the factors of the given clues? There are plenty of places to be tricked in this puzzle, so good luck!

Print the puzzles or type the solution in this excel file: 12 factors 1063-1072

Now I’ll write a little bit about the number 1069:

  • 1069 is a prime number.
  • Prime factorization: 1069 is prime.
  • The exponent of prime number 1069 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1069 has exactly 2 factors.
  • Factors of 1069: 1, 1069
  • Factor pairs: 1069 = 1 × 1069
  • 1069 has no square factors that allow its square root to be simplified. √1069 ≈ 32.695565

How do we know that 1069 is a prime number? If 1069 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1069 ≈ 32.7. Since 1069 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1069 is a prime number.

The next prime number won’t be until 1087, making 1069 have a prime gap of 18. That’s two numbers short of the record for primes up to 1087, so it’s quite a lot!

Not only is 1069 a prime number, but OEIS.org informs us that
310693 is a prime number,
33106933 is a prime number,
3331069333 is a prime number, and
333310693333 is also a prime number!

30² + 13² = 1069

1069 is the hypotenuse of a Pythagorean triple:
731-780-1069 calculated from 30² – 13², 2(30)(13), 30² + 13²

Here’s another way we know that 1069 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 30² + 13² = 1069 with 30 and 13 having no common prime factors, 1069 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1069 ≈ 32.7. Since 1069 is not divisible by 5, 13, 17, or 29, we know that 1069 is a prime number.

1069 looks interesting when it is written in a few other bases:
It’s palindrome 565 in BASE 14 because 5(14²) + 6(14) + 5(1) = 1069, and
4B4 in BASE 15 (B is 11 base 10) because 4(15²) + 11(15) + 4(1) = 1069, and
it’s consecutive odd digits 357 in BASE 18 because 3(18²) + 5(18) + 7(1) = 1069

1068 and Level 6

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The permissible common factors of 40 and 48 are 4 and 8, and the permissible common factors of 54 and 36 are 6 and 9. By themselves, neither of those pairs of clues is enough to get you started with this level 6 puzzle. You will have to study ALL the clues to begin to solve it. Believe it or not, sometimes a row or column with no clue is the best place to start.

Print the puzzles or type the solution in this excel file: 12 factors 1063-1072

This is my 1068th post so let me share a little information about the number 1068:

  • 1068 is a composite number.
  • Prime factorization: 1068 = 2 × 2 × 3 × 89, which can be written 1068 = 2² × 3 × 89
  • 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 1068 has exactly 12 factors.
  • Factors of 1068: 1, 2, 3, 4, 6, 12, 89, 178, 267, 356, 534, 1068
  • Factor pairs: 1068 = 1 × 1068, 2 × 534, 3 × 356, 4 × 267, 6 × 178, or 12 × 89,
  • Taking the factor pair with the largest square number factor, we get √1068 = (√4)(√267) = 2√267 ≈ 32.68027

1068 is the hypotenuse of a Pythagorean triple:
468-960-1068 which is 12 times (39-80-89)

1067 and Level 5

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The common factors of 20 and 40 are 1, 2, 4, 5, and 10. Only the ones in blue will put numbers from 1 to 12 in the top row, as required. Since there is more than one possible common factor, don’t start with those two clues. This is a level 5 puzzle so at least one pair of clues will work to get you started.

Print the puzzles or type the solution in this excel file: 12 factors 1063-1072

Here are a few facts about the number 1067:

  • 1067 is a composite number.
  • Prime factorization: 1067 = 11 × 97
  • 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 1067 has exactly 4 factors.
  • Factors of 1067: 1, 11, 97, 1067
  • Factor pairs: 1067 = 1 × 1067 or 11 × 97
  • 1067 has no square factors that allow its square root to be simplified. √1067 ≈ 32.66497

1067 is the hypotenuse of a Pythagorean triple:
715-792-1067 which is 11 times (65-72-97)
We can use the 11 divisibility trick on all the numbers in that triple:
7 – 1 + 5 = 11
7 – 9 + 2 = 0
1 – 0 + 6 – 7 = 0
to see that all of them can indeed be evenly divided by 11.

1067 is palindrome 1F1 in BASE 26 (F is 15 base 10) because 26² + 15(26) + 1 = 1067

1066 and Level 4

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Some of the clues in today’s puzzle were used in previous puzzles this week. Sometimes their factors have to be exactly the same as they were in the previous puzzles, but sometimes they might not be. Can you figure out where to put the numbers 1 to 12 in both the first column and the top row so that those numbers are the factors of the clues given?

Print the puzzles or type the solution in this excel file: 12 factors 1063-1072

Here are a few facts about the number 1066:

  • 1066 is a composite number.
  • Prime factorization: 1066 = 2 × 13 × 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 1066 has exactly 8 factors.
  • Factors of 1066: 1, 2, 13, 26, 41, 82, 533, 1066
  • Factor pairs: 1066 = 1 × 1066, 2 × 533, 13 × 82, or 26 × 41
  • 1066 has no square factors that allow its square root to be simplified. √1066 ≈ 32.649655

29² + 15² = 1066
25² + 21² = 1066

1066 is the hypotenuse of FOUR Pythagorean triples:
616-870-1066 is 2 times (308-435-533) and calculated from 29² – 15², 2(29)(15), 29² + 15²
410-984-1066 which is (5-12-13) times 82
234-1040-1066 which is 26 times (9-40-41)
184-1050-1066 is 2 times (92-525-533) and calculated from 25² – 21², 2(25)(21), 25² + 21²

1066 looks interesting when it is written in some other bases:
It’s palindrome 1110111 in BASE 3 because 3⁶ + 3⁵ + 3⁴ + 3² + 3¹ +3⁰ = 1066,
13231 in BASE 5 because 1(5⁴) + 3(5³) + 2(5²) + 3(5) + 1(1) = 1066,
1414 in BASE 9 because 1(9³) + 4(9²) + 1(9) + 4(1) = 1066, and
2I2 in BASE 19 (I is 18 base 10) because 2(19²) + 18(19) + 2(1) = 1066

 

 

1065 and Level 3

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24 and 14 have two common factors, but just one of them will put only numbers from 1 to 12 in the first column. After you write those factors on the puzzle, work down the first column of the puzzle cell by cell writing the appropriate factors as you go.

Print the puzzles or type the solution in this excel file: 12 factors 1063-1072

Here are a few facts about the number 1065:

  • 1065 is a composite number.
  • Prime factorization: 1065 = 3 × 5 × 71
  • 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 1065 has exactly 8 factors.
  • Factors of 1065: 1, 3, 5, 15, 71, 213, 355, 1065
  • Factor pairs: 1065 = 1 × 1065, 3 × 355, 5 × 213, or 15 × 71
  • 1065 has no square factors that allow its square root to be simplified. √1065 ≈ 32.63434

1065 is the hypotenuse of a Pythagorean triple:
639-852-1065 which is (3-4-5) times 213

1065 is a palindrome in two bases:
It’s 353 in BASE 18 because 3(18²) + 5(18) + 3(1) = 1065
1A1 in BASE 28 (A is 10 base 10) because 28² + 10(28) + 1 = 1065

1064 and Level 2

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If you did yesterday’s puzzle, then you will recognize four of the clues in today’s puzzle. They will give you a good start in finding all the rest of factors. See how well you do on this one!

Print the puzzles or type the solution in this excel file: 12 factors 1063-1072

Now I’ll tell you a little bit about the number 1064:

Its 0 is an even digit, and its last two digits, 64, can be evenly divided by 8, so 1064 is also divisible by 8.

  • 1064 is a composite number.
  • Prime factorization: 1064 = 2 × 2 × 2 × 7 × 19, which can be written 1064 = 2³ × 7 × 19
  • 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 1064 has exactly 16 factors.
  • Factors of 1064: 1, 2, 4, 7, 8, 14, 19, 28, 38, 56, 76, 133, 152, 266, 532, 1064
  • Factor pairs: 1064 = 1 × 1064, 2 × 532, 4 × 266, 7 × 152, 8 × 133, 14 × 76, 19 × 56, or 28 × 38
  • Taking the factor pair with the largest square number factor, we get √1064 = (√4)(√266) = 2√266 ≈ 32.61901

The difference in the numbers in one of its factor pairs, 28 × 38, is exactly ten, so we are exactly 25 away from the next perfect square number:
33² – 5² = 1089 – 25 = 1064

I like the way 1064 looks in a couple of other bases:
It’s 888 in BASE 11 because 8(11² + 11 + 1) = 8(133) = 1064, and
it’s 248 in BASE 22 because 2(22²) + 4(22) + 8(1) = 1064

1062 Complicated Logic

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The logic used to solve this particular level 6 puzzle is complicated, but answering the following questions in the order given will help you to see and understand that logic:

  1. Two of the numbers from 1 to 10 have only one clue each in this puzzle. What are those two numbers? The product of those two numbers is the missing clue that you will use later in the puzzle.
  2. Which two clues MUST use both 1’s?
  3. Which four clues must use all the 3’s and 6’s?
  4. Can both 30’s be 3 × 10 or be 5 × 6?
  5. Can both 40’s be 4 × 10 or be 5 × 8?
  6. What MUST be the factors of 24 in this puzzle?
  7. What clues must use both 4’s? What clues must use both 8s?
  8. Is 1 or 2 the common factor for clues 8 and 2 that will make the puzzle work?
  9. Is 5 or 10 the common factor for clues 30 and 40 near the bottom of the puzzle?

Once you know the answers to those questions and the two sets of common factors, you can very quickly complete the puzzle.

Print the puzzles or type the solution in this excel file: 10-factors-1054-1062

Here’s some information about the number 1062:

  • 1062 is a composite number.
  • Prime factorization: 1062 = 2 × 3 × 3 × 59, which can be written 1062 = 2 × 3² × 59
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1062 has exactly 12 factors.
  • Factors of 1062: 1, 2, 3, 6, 9, 18, 59, 118, 177, 354, 531, 1062
  • Factor pairs: 1062 = 1 × 1062, 2 × 531, 3 × 354, 6 × 177, 9 × 118, or 18 × 59,
  • Taking the factor pair with the largest square number factor, we get √1062 = (√9)(√118) = 3√118 ≈ 32.58834

1062 looks interesting when it is written in a couple of different bases:
It’s 2D2 in BASE 20 (D is 13 base 10) because 2(20²) + 13(20) + 2(1) = 1062
and 246 in BASE 22 because 2(22²) + 4(22) + 6(1) = 1062

1060 A Challenge for Justin’s Birthday

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Justin had no problems solving the puzzle I made for his last birthday, so this year I’ve made it tougher. Write the numbers from 1 to 10 in each of the four purple sections of the puzzle so that the clues are the products of the corresponding factors. Like always, there is only one solution. This puzzle can still be solved entirely by using logic and knowledge of basic multiplication and division facts. Will Justin be able to figure it out? I’m anxious to find out. Happy birthday, Justin!

Here’s the same puzzle without added color if that’s better for printing.

Print the puzzles or type the solution in this excel file: 10-factors-1054-1062

Now here are some things I’ve learned about the number 1060:

  • 1060 is a composite number.
  • Prime factorization: 1060 = 2 × 2 × 5 × 53, which can be written 1060 = 2² × 5 × 53
  • 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 1060 has exactly 12 factors.
  • Factors of 1060: 1, 2, 4, 5, 10, 20, 53, 106, 212, 265, 530, 1060
  • Factor pairs: 1060 = 1 × 1060, 2 × 530, 4 × 265, 5 × 212, 10 × 106, or 20 × 53,
  • Taking the factor pair with the largest square number factor, we get √1060 = (√4)(√265) = 2√265 ≈ 32.55764

1060 is the sum of consecutive prime numbers three different ways:
It is the sum of the prime numbers from 2 to 97, that’s all the prime numbers less than 100.
83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 = 1060; that’s 10 consecutive primes
257 + 263 + 269 + 277 = 1060; that’s 4 consecutive primes

32² + 6² = 1060
24² + 22² = 1060

1060 is the hypotenuse of FOUR Pythagorean triples:
92-1056-1060
560-900-1060
636-848-1060
384-988-1060

1060 is a palindrome in some other bases:
884 in BASE 11
424 in BASE 16
202 in BASE 23

1059 and Level 4

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A level four puzzle is only a little more difficult than a level three puzzle. Instead of starting with the top clue and working down cell by cell, the next clue that you need could be anywhere in the puzzle. It may be a little harder, but you can still solve this puzzle!

Print the puzzles or type the solution in this excel file: 10-factors-1054-1062

What can I tell you about the number 1059?

  • 1059 is a composite number.
  • Prime factorization: 1059 = 3 × 353
  • 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 1059 has exactly 4 factors.
  • Factors of 1059: 1, 3, 353, 1059
  • Factor pairs: 1059 = 1 × 1059 or 3 × 353
  • 1059 has no square factors that allow its square root to be simplified. √1059 ≈ 32.54228

1059 is the hypotenuse of a Pythagorean triple:
675-816-1059 which is 3 times (225-272-353)

1059 is palindrome 636 in BASE 13 because 6(13²) + 3(13) + 6(1) = 1059