1133 A Challenge Puzzle

I haven’t published a Find the Factors 1 – 10 Challenge puzzle for a while. Getting started on this one shouldn’t be difficult. The challenge will be in finishing it! Don’t guess and check. It can all be done using logic.

Print the puzzles or type the solution in this excel file: 10-factors-1121-1133

Here are some facts about the number 1133:

Since both 11 and 33 are divisible by 11, we know 1133 can be evenly divided by 11.

  • 1133 is a composite number.
  • Prime factorization: 1133 = 11 × 103
  • 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 1133 has exactly 4 factors.
  • Factors of 1133: 1, 11, 103, 1133
  • Factor pairs: 1133 = 1 × 1133 or 11 × 103
  • 1133 has no square factors that allow its square root to be simplified. √1133 ≈ 33.66007

1133 is the sum of the twenty-one prime numbers from 13 to 101.
1133 is also the sum of the fifteen prime numbers from 43 to 107.

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1072 Find the Factors Challenge

These Find the Factors Challenge puzzles are tougher than my other puzzles, but they can still be solved using logic and basic multiplication and division facts. Go ahead, give it a try! I sincerely hope you will succeed!

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

Now I’ll share some facts about the number 1072:

  • 1072 is a composite number.
  • Prime factorization: 1072 = 2 × 2 × 2 × 2 × 67, which can be written 1072 = 2⁴ × 67
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1072 has exactly 10 factors.
  • Factors of 1072: 1, 2, 4, 8, 16, 67, 134, 268, 536, 1072
  • Factor pairs: 1072 = 1 × 1072, 2 × 536, 4 × 268, 8 × 134, or 16 × 67
  • Taking the factor pair with the largest square number factor, we get √1072 = (√16)(√67) = 4√67 ≈ 32.74141

1072 is also palindrome 646 in BASE 13 because 6(13²) + 4(13) + 6(1) = 1072

1060 A Challenge for Justin’s Birthday

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

1053 Find the Factors Challenge

If you can multiply and divide and THINK, then you can solve this puzzle. Go ahead. Give it a try! To solve it, write the numbers from 1 to 10 in each of the four bold areas on the puzzle so that the given clues will be the products of those corresponding factors. Good luck!

Print the puzzles or type the solution in this excel file: 12 factors 1044-1053

What can I tell you about the number 1053?

  • 1053 is a composite number.
  • Prime factorization: 1053 = 3 × 3 × 3 × 3 × 13, which can be written 1053 = 3⁴ × 13
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1053 has exactly 10 factors.
  • Factors of 1053: 1, 3, 9, 13, 27, 39, 81, 117, 351, 1053
  • Factor pairs: 1053 = 1 × 1053, 3 × 351, 9 × 117, 13 × 81, or 27 × 39
  • Taking the factor pair with the largest square number factor, we get √1053 = (√81)(√13) = 9√13 ≈ 32.44996

27² + 18²  = 1053 so 1053 is the hypotenuse of a Pythagorean triple:
405-972-1053 calculated from 27² – 18², 2(27)(18), 27² + 18²

1053 is the sum of three consecutive powers of 3:
3 + 3 + 3 = 1053

1053 is 3033 in BASE 7 because 3(7³ + 7¹ + 7º) = 3(351) = 1053, and
it’s palindrome 878 in BASE 11 because 8(121) + 7(11) + 8(1) = 1053

1043 Find the Factors Challenge Puzzle

I made this particular Find the Factors 1-10 Challenge puzzle three weeks ago. It took me just under 30 minutes to solve it when I tried it again before publishing it. How long will it take you to solve it?

Print the puzzles or type the solution in this excel file: 10-factors-1035-1043

Now here’s a little about the number 1043:

  • 1043 is a composite number.
  • Prime factorization: 1043 = 7 × 149
  • 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 1043 has exactly 4 factors.
  • Factors of 1043: 1, 7, 149, 1043
  • Factor pairs: 1043 = 1 × 1043 or 7 × 149
  • 1043 has no square factors that allow its square root to be simplified. √1043 ≈ 32.29551

1043 is the sum of consecutive prime numbers two different ways:
It’s the sum of the 21 prime numbers from 11 to 97 and,
it’s the sum of the 13 prime numbers from 53 to 107.

1043 is also the hypotenuse of a Pythagorean triple:
357-980-1043 which is 7 times (51-140-149)

 

1034 Find the Factors Challenge Puzzle

One of the things I really like about these Challenge puzzles is that you have to use logic. Guessing and checking just won’t work. Go ahead and give this puzzle a try!

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

Here are a few facts about the number 1034:

1 – 0 + 3 – 4 = 0, so 1034 can be evenly divided by 11.

Because 1034 = 2(47)(11), it is the short leg of a few Pythagorean triples including
1034-2088-2330 calculated from 2(47)(11), 47² – 11², 47² + 11²

  • 1034 is a composite number.
  • Prime factorization: 1034 = 2 × 11 × 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 1034 has exactly 8 factors.
  • Factors of 1034: 1, 2, 11, 22, 47, 94, 517, 1034
  • Factor pairs: 1034 = 1 × 1034, 2 × 517, 11 × 94, or 22 × 47
  • 1034 has no square factors that allow its square root to be simplified. √1034 ≈ 32.15587

1027 Find the Factors Challenge

I’m really enjoying these Find the Factors Challenge puzzles, and I hope that you will give them a try and love them, too. You can find this one as well as a little less challenging one in the excel file link below the puzzle. You can type the factors directly on that file. Remember to use logic for every single factor pair you use.

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

Now I’ll tell you a little about the number 1027:

It is the sum of seven consecutive prime numbers:
131 + 137 + 139 + 149 + 151 + 157 + 163 = 1027

It is the sum of the squares of the first eight prime numbers:
2² +  3² +  5² +  7² +  11² +  13² +  17² +  19² = 1027
Indeed, 666 + 19² = 1027. Thanks to Stetson.edu for that fun fact.

Because 19³ – 18³ = 1027, it is the 19th Centered Hexagonal Number.
That also means that 19² + 19(18) + 18² = 1027
because a³ – b³ = (a-b)(a²+ab+b²).

1027 is the hypotenuse of a Pythagorean triple:
395-948-1027 which is (5-12-13) times 79.

1027 is also a palindrome when it is written in these three other bases:
717 in BASE 12
535 in BASE 14
1B1 in BASE 27 (B is 11 base 10)

  • 1027 is a composite number.
  • Prime factorization: 1027 = 13 × 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 1027 has exactly 4 factors.
  • Factors of 1027: 1, 13, 79, 1027
  • Factor pairs: 1027 = 1 × 1027 or 13 × 79
  • 1027 has no square factors that allow its square root to be simplified. √1027 ≈ 32.0468

 

1019 An Easier Find the Factors Challenge Puzzle

I’ve recently posted some more challenging puzzles that I’ve named Find the Factors 1 – 10 Challenge, and they definitely are a more challenging puzzle than one of my more traditional level 6 puzzles. As of today, no one has informed me that they have been able to solve either puzzle number 1000 or 1010.

Two years ago I made perhaps my most challenging level 6 puzzle, a 16 × 16 puzzle to commemorate Steve Morris’s birthday. Steve Morris was the very first person to type a comment on my blog, and I have appreciated his encouragement over the years. Steve has solved many kinds of puzzles in his life including some of the toughest I have made, but the puzzle I made for that birthday was no picnic for even him to complete.

This year I’ve made him a challenging puzzle, but it is still a little easier than the other two challenge puzzles I’ve made. If you’ve tried either of those other puzzles without success, still give this one a try. Good luck to you all, and Happy Birthday to Steve Morris! I saved this post number (1019) for you because it uses your birthdate numbers, howbeit out of order.

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

This is my 1019th post. Here are a few facts about the number 1019.

Prime number 1019 is the sum of the 19 prime numbers from 17 to 97.

  • 1019 is a prime number.
  • Prime factorization: 1019 is prime.
  • The exponent of prime number 1019 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1019 has exactly 2 factors.
  • Factors of 1019: 1, 1019
  • Factor pairs: 1019 = 1 × 1019
  • 1019 has no square factors that allow its square root to be simplified. √1019 ≈ 31.921779

How do we know that 1019 is a prime number? If 1019 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1019 ≈ 31.9. Since 1019 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1019 is a prime number.

1010 Find the Factors Challenge

Level 6 puzzles are difficult. At least they are until you’ve done a few, then they become much less difficult to solve.

This Find the Factors 1 – 10 Challenge is much more difficult. I published the first of these puzzles (#1000) last week, but I won’t necessarily make one every week. If you’ve done a few level 6 puzzles, see how you do with a Challenge Puzzle.

Print the puzzles or type the solution in this excel file: 10-factors-1002-1011

Here’s a little about the number 1010:

1010 is the sum of two squares two different ways:
29² + 13² = 1010
31² + 7² = 1010

That means that 1010 is the hypotenuse of some Pythagorean triples:
200-990-1010 which is 10 times (20-99-101)
434-912-1010 calculated from 2(31)(7), 31² – 7², 31² + 7²
606-808-1010 which is (3-4-5) times 202
672-754-1010 calculated from 29² – 13², 2(29)(13), 29² + 13²

1010 is a fun-looking number in base 10.
It is also palindrome 262 in BASE 21 because 2(21²) + 6(21) + 2(1) = 1010

  • 1010 is a composite number.
  • Prime factorization: 1010 = 2 × 5 × 101
  • 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 1010 has exactly 8 factors.
  • Factors of 1010: 1, 2, 5, 10, 101, 202, 505, 1010
  • Factor pairs: 1010 = 1 × 1010, 2 × 505, 5 × 202, or 10 × 101
  • 1010 has no square factors that allow its square root to be simplified. √1010 ≈ 31.780497

1000 Find the Factors Challenge

Dan Bach asked me if I had anything special planned for my 1000 post. I had a few things planned, but this Find the Factors Challenge exists because of his question and our twitter conversation. I like this puzzle better than my previous plans. It is only fitting that I dedicate this puzzle to Dan Bach.

There are four places outlined in bold that need you to place the numbers 1 – 10 so that each section of the puzzle will work as a multiplication table. Each section of the puzzle is dependent on the other sections. None of the sections could be a puzzle independent of the others. You will use similar logic to solve this puzzle as you would a regular Find the Factors puzzle. It will not be easy to solve at all, but give it a try. There is only one solution.

Print the puzzles or type the solution in this updated excel file: 12 factors 993-1001

1000 is a perfect cube as well as the product of perfect cubes.

30² + 10² = 1000
26² + 18² = 1000

1000 is the hypotenuse of three Pythagorean triples:
600² + 800² = 1000²
352² + 936² = 1000²
280² + 960² = 1000²

1000 looks interesting in a few bases:
2626 in BASE 7
1331 in BASE 9
1A1 in BASE 27 (A is 10 base 10)

  • 1000 is a composite number.
  • Prime factorization: 1000 = 2 × 2 × 2 × 5 × 5 × 5, which can be written 1000 = 2³ × 5³
  • The exponents in the prime factorization are 3 and 3. Adding one to each and multiplying we get (3 + 1)(3 + 1) = 4 × 4 = 16. Therefore 1000 has exactly 16 factors.
  • Factors of 1000: 1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, 125, 200, 250, 500, 1000
  • Factor pairs: 1000 = 1 × 1000, 2 × 500, 4 × 250, 5 × 200, 8 × 125, 10 × 100, 20 × 50, or 25 × 40
  • Taking the factor pair with the largest square number factor, we get √1000 = (√100)(√10) = 10√10 ≈ 31.622777