Factors

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

1058 and Level 3

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You can solve this puzzle! Just start with the easy clues closest to the top and work your way down cell by cell until you have the numbers from 1 to 10 in both the first column and the top row. Have fun!

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

Now I’ll share some information about the number 1058:

  • 1058 is a composite number.
  • Prime factorization: 1058 = 2 × 23 × 23, which can be written 1058 = 2 × 23²
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 × 3  = 6. Therefore 1058 has exactly 6 factors.
  • Factors of 1058: 1, 2, 23, 46, 529, 1058
  • Factor pairs: 1058 = 1 × 1058, 2 × 529, or 23 × 46
  • Taking the factor pair with the largest square number factor, we get √1058 = (√529)(√2) = 23√2 ≈ 32.52691

1057 and Level 2

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Can you figure out where to put the numbers from 1 to 10 in both the 1st column and the top row so that this puzzle behaves like a multiplication table?

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

Here’s a little bit about the number 1057:

  • 1057 is a composite number.
  • Prime factorization: 1057 = 7 × 151
  • 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 1057 has exactly 4 factors.
  • Factors of 1057: 1, 7, 151, 1057
  • Factor pairs: 1057 = 1 × 1057 or 7 × 151
  • 1057 has no square factors that allow its square root to be simplified. √1057 ≈ 32.5115

1056 How to Tile a 32 × 33 Floor

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32 × 33 = 1056, and OEIS.org informs us that those are the smallest rectangular dimensions that can be tiled with different perfect squares.

It isn’t difficult to do the tiling. All you have to remember is 32 × 33 and to put an 18 × 18 tile in a corner. The rest of the perfect square tiles seem to almost fall into place as this gif I made illustrates:

Tiling a 32 × 33 Rectangle

make science GIFs like this at MakeaGif
Now I’ll tell you a little about the number 1056:
  • 1056 is a composite number.
  • Prime factorization: 1056 = 2 × 2 × 2 × 2 × 2 × 3 × 11, which can be written 1056 = 2⁵ × 3 × 11
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 6 × 2 × 2 = 24. Therefore 1056 has exactly 24 factors.
  • Factors of 1056: 1, 2, 3, 4, 6, 8, 11, 12, 16, 22, 24, 32, 33, 44, 48, 66, 88, 96, 132, 176, 264, 352, 528, 1056
  • Factor pairs: 1056 = 1 × 1056, 2 × 528, 3 × 352, 4 × 264, 6 × 176, 8 × 132, 11 × 96, 12 × 88, 16 × 66, 22 × 48, 24 × 44, or 32 × 33
  • Taking the factor pair with the largest square number factor, we get √1056 = (√16)(√66) = 4√66 ≈ 32.49615
Since 1056 is the product of consecutive numbers, 32 × 33, it is the sum of the first 32 even numbers:
2 + 4 + 6 + 8 + 10 + . . .  +56 + 58 + 60 + 62 + 64 = 1056

 

1055 and Level 1

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I bet you can solve this puzzle like clockwork!

 

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

Here is a little bit about the number 1055:

  • 1055 is a composite number.
  • Prime factorization: 1055 = 5 × 211
  • 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 1055 has exactly 4 factors.
  • Factors of 1055: 1, 5, 211, 1055
  • Factor pairs: 1055 = 1 × 1055 or 5 × 211
  • 1055 has no square factors that allow its square root to be simplified. √1055 ≈ 32.48076

How Can You Count These 1054 Tiny Squares?

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There are 1054 tiny squares in the image below, making 1054 a centered triangular number. How can you know that I’m not pulling the wool over your eyes about the number of tiny squares? Here are a few ways that you can quickly count all of them.

If you start with the yellow square in the center and count outward each succeeding triangle you will get 1 yellow square + 3 green squares + 6 blue squares + 9 purple squares + 12 red squares + 15 orange squares, etc. until you reach the final 78 blue squares:
1 + 3 + 6 + 9 + 12 + 15 + . . . + 78
= 1 + 3(1 + 2 + 3 + 4 + 5 + . . . + 26)
= 1 + 3(26*27)/2 = 1 + 3(351) = 1054

Using a little bit of algebra, you can show that
1 + 3(26*27)/2 = (3(26²) + 3(26) + 2)/2 = 1054

You can divide the centered triangle above into three triangles as I also did in the graphic. The three triangles represent the 25th, the 26th, and the 27th triangular numbers. Adding them up you get:
25(26)/2 + 26(27)/2 + 27(28)/2 = 325 + 351 + 378 = 1054

Here is some more information about the number 1054:

  • 1054 is a composite number.
  • Prime factorization: 1054 = 2 × 17 × 31
  • 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 1054 has exactly 8 factors.
  • Factors of 1054: 1, 2, 17, 31, 34, 62, 527, 1054
  • Factor pairs: 1054 = 1 × 1054, 2 × 527, 17 × 62, or 31 × 34
  • 1054 has no square factors that allow its square root to be simplified. √1054 ≈ 32.465366

1054 is the sum of six consecutive prime numbers:
163 + 167 + 173 + 179 + 181 + 191 = 1054

1054 is the hypotenuse of a Pythagorean triple:
496-930-1054 which is (8-15-17) times 62

1054 looks interesting when it is written in some other bases:
It’s 4A4 in BASE 15 (A is 10 base 10) because 4(15²) + 10(15) + 4(1) = 1054
1C1 in BASE 27 (C is 12 base 10) because 27² + 12(27) + 1 = 1054
VV in BASE 33 (V is 31 base 10) because 31(33) + 31(1) = 31(34) = 1054
V0 in BASE 34 because 31(34) = 1054

 

 

 

 

1053 Find the Factors Challenge

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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:
3⁴ + 3⁵ + 3⁶ = 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

1052 A Mysterious Purple Cat for Josephine

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My very brilliant friend, Josephine likes cats and her favorite color is purple. Hence, this is a purple cat puzzle made especially for her. It’s a mystery level puzzle so its difficulty level is a big secret. This puzzle only requires skills in multiplication and division. I’m sure it will be no match for Josephine, who can easily handle more advanced mathematics such as calculus. Josephine is also fluent in English, Chinese, Spanish, French, Arabic, and Tajiki.  She is very busy, so hopefully, she’ll be able to find the time to spend some time with this mysterious purple cat.

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

If the colors in the puzzle distract you, here is the same puzzle in very plain black and white:

Here is some information about the number 1052:

Its last two digits are 52, so it is divisible by 4.

  • 1052 is a composite number.
  • Prime factorization: 1052 = 2 × 2 × 263, which can be written 1052 = 2² × 263
  • 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 1052 has exactly 6 factors.
  • Factors of 1052: 1, 2, 4, 263, 526, 1052
  • Factor pairs: 1052 = 1 × 1052, 2 × 526, or 4 × 263
  • Taking the factor pair with the largest square number factor, we get √1052 = (√4)(√263) = 2√263 ≈ 32.434549480453709548485

1052 is palindrome 282 in BASE 21 because 2(21²) + 8(21) + 2(1) = 1052