1084 and Level 4

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Where should you put all the numbers 1 to 12 in both the top row and the first column?  You will have to think about it and use logic. Some of the clues might be tricky, but you’ll figure it all out.

Print the puzzles or type the solution in this excel file: 12 factors 1080-1086

Here are some facts about the number 1084:

  • 1084 is a composite number.
  • Prime factorization: 1084 = 2 × 2 × 271, which can be written 1084 = 2² × 271
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (2 + 1)(1 + 1) = 3 × 2  = 6. Therefore 1084 has exactly 6 factors.
  • Factors of 1084: 1, 2, 4, 271, 542, 1084
  • Factor pairs: 1084 = 1 × 1084, 2 × 542, or 4 × 271
  • Taking the factor pair with the largest square number factor, we get √1084 = (√4)(√271) = 2√271 ≈ 32.92416

1084 is the sum of the 18 prime numbers from 23 to 101.

It is also the sum of six consecutive prime numbers:
167 + 173 + 179 + 181 + 191 + 193  = 1084

1083 and Level 3

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Start with the two clues near the top of this level 3 puzzle. Find their common factor that will put only numbers from 1 to 12 in the top row. Then work down the puzzle row by row filling in factors from 1 to 12 as you go. It won’t take you long to complete this puzzle!

Print the puzzles or type the solution in this excel file: 12 factors 1080-1086

1 + 0 + 8 + 3 = 12, so 1083 can be evenly divided by 3. What else can I tell you about that number?

  • 1083 is a composite number.
  • Prime factorization: 1083 = 3 × 19 × 19, which can be written 1083 = 3 × 19²
  • 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 1083 has exactly 6 factors.
  • Factors of 1083: 1, 3, 19, 57, 361, 1083
  • Factor pairs: 1083 = 1 × 1083, 3 × 361, or 19 × 57
  • Taking the factor pair with the largest square number factor, we get √1083 = (√361)(√3) = 19√3 ≈ 32.90897

1083 looks interesting when it is written in some other bases:
It’s 575 in BASE 14 because 5(14²) + 7(14) + 5(1) = 1083,
363 in BASE 18 because 3(18²) + 6(18) + 3(1) = 3(18² + 36 + 1) = 3(361) = 1083,
300 in BASE 19 because 3(19²) = 1083, and
it’s 212 in BASE 23 because 2(23²) + 1(23) + 2(1) = 1083

1082 and Level 3

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Write the numbers from 1 to 12 in both the first column and the top row so that this puzzle can function as a multiplication table. Do that, and you will have found all the factors 1-12 and solved the puzzle. Afterward, you or someone else can also complete the entire table, if you’d like.

Print the puzzles or type the solution in this excel file: 12 factors 1080-1086

Here is a little about the number 1082:

  • 1082 is a composite number.
  • Prime factorization: 1082 = 2 × 541
  • 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 1082 has exactly 4 factors.
  • Factors of 1082: 1, 2, 541, 1082
  • Factor pairs: 1082 = 1 × 1082 or 2 × 541
  • 1082 has no square factors that allow its square root to be simplified. √1082 ≈ 32.89377

31² + 11² = 1082

1082 is the hypotenuse of a Pythagorean triple:
682-840-1082 calculated from 2(31)(11), 31² – 11², 31² + 11²,
It is also 2 times (341-420-541)  or 2(21² – 10²), 4(21)(10), 2(21² + 10²)

1082 is palindrome 2E2 in BASE 20 (E is 14 base 10)
because 2(20²) + 14(20) + 2(1) = 1082

1081 and Level 1

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If you know basic division and multiplication facts for factors 1 to 12, then you can complete this whole puzzle and make it be a multiplication table but with the factors not in their usual places.

Print the puzzles or type the solution in this excel file: 12 factors 1080-1086

Here are a few facts about the number 1081:

  • 1081 is a composite number.
  • Prime factorization: 1081 = 23 × 47
  • 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 1081 has exactly 4 factors.
  • Factors of 1081: 1, 23, 47, 1081
  • Factor pairs: 1081 = 1 × 1081 or 23 × 47
  • 1081 has no square factors that allow its square root to be simplified. √1081 ≈ 32.87856

1081 is the 46th triangular number because 46(47)/2 = 1081
That means that the sum of the numbers from 1 to 46 is 1081:
1 + 2 + 3 + 4 + . . . + 43 + 44 + 45 + 46 = 1081

1081 is also the sum of eleven consecutive prime numbers:
73 + 79 + 83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 = 1081

In other bases, 1081 is 3 different palindromes that begin and end with 1:
1L1 in BASE 24 (L is 21 base 10) because 24² + 21(24) + 1 = 1081
1D1 in BASE 27 (D is 13 base 10) because 27² + 13(27) + 1 = 1081
161 in BASE 30 because 30² + 6(30) + 1 = 1081

STOP! Look How Cool a Number 1080 Is!

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STOP Sign Geometry Fact:

What can I tell you about the number 1080?  Lots of things!

The sum of the interior angles of an eight-sided polygon such as a stop sign is 1080°.

The sum of the interior angles of an octagon is 6(180°) = 1080°. Convex or Concave, it doesn’t matter, the sum of those interior angles of an eight-sided polygon will still be 1080°, as illustrated below:

Factor Trees for 1080:

Here are a couple of the MANY possible factor trees for 1080:

Factors of 1080:

There is only one number less than 1080 that has as many factors as 1080 does. What was that number? 840. How many factors does1080 have? 32. Wow!

  • 1080 is a composite number.
  • Prime factorization: 1080 = 2 × 2 × 2 × 3 × 3 × 3 × 5, which can be written 1080 = 2³ × 3³ × 5.
  • The exponents in the prime factorization are 3, 3, and 1. Adding one to each and multiplying we get (3 + 1)(3 + 1)(1 + 1) ) = 4 × 4 × 2 = 32. Therefore 1080 has exactly 32 factors.
  • Factors of 1080: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 27, 30, 36, 40, 45, 54, 60, 72, 90, 108, 120, 135, 180, 216, 270, 360, 540, 1080
  • Factor pairs: 1080 = 1 × 1080, 2 × 540, 3 × 360, 4 × 270, 5 × 216, 6 × 180, 8 × 135, 9 × 120, 10 × 108, 12 × 90, 15 × 72, 18 × 60, 20 × 54, 24 × 45, 27 × 40 or 30 × 36
  • Taking the factor pair with the largest square number factor, we get √1080 = (√36)(√30) = 6√30 ≈ 32.86335

1080 has 28 composite factors and is the smallest number that can make that claim. (Of its 32 factors, all are composite numbers except 1, 2, 3, and 5). That’s more than 840’s 27 composite factors. (Its 32 factors minus 1, 2, 3, 5, and 7)

Sum Difference Puzzles:

30 has four factor pairs. One of those pairs adds up to 13, and  another one subtracts to 13. Put the factors in the appropriate boxes in the first puzzle.

1080 has sixteen factor pairs. One of the factor pairs adds up to ­78, and a different one subtracts to 78. If you can identify those factor pairs, then you can solve the second puzzle!

The second puzzle is really just the first puzzle in disguise. Why would I say that?

Other Facts about the Number 1080:

1080 is the sum of these four consecutive prime numbers:
263 + 269 + 271 + 277 = 1080

1080 is the sum of four consecutive powers of three:
3⁶ + 3⁵ + 3⁴ + 3³ = 1080

1080 is the hypotenuse of a Pythagorean triple:
648-864-1080 which is (3-4-5) times 216

Note that 5(6³) = 5(216) = 1080 so 1080 is 500 in BASE 6.
It’s palindrome 252 in BASE 22 because 2(22²) + 5(22) + 2(1) = 1080,
UU in BASE 35 (U is 30 base 10) because 30(35) + 30(1) = 30(36) = 1080,
and it’s U0 in BASE 36 because 30(36) = 1080

And now I’ll STOP writing about how cool 1080 is.

1079 An Easier Find The Factors Challenge?

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This find the factors challenge puzzle might be easier than most of the challenge puzzles are, but it will still give you plenty of reasons to think about what factors you should put where. You need to put all the numbers from 1 to 10 in each of the four bold areas so that those numbers are the factors of the given clues. There is only one solution. Can you find it?

Print the puzzles or type the solution in this excel file: 10-factors-1073-1079

Here’s a little about the number 1079:

  • 1079 is a composite number.
  • Prime factorization: 1079 = 13 × 83
  • 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 1079 has exactly 4 factors.
  • Factors of 1079: 1, 13, 83, 1079
  • Factor pairs: 1079 = 1 × 1079 or 13 × 83
  • 1079 has no square factors that allow its square root to be simplified. √1079 ≈ 32.84814

1079 is the sum of three consecutive prime numbers:
353 + 359 + 367 = 1079

1079 is also the hypotenuse of a Pythagorean triple:
415-996-1079 which is (5-12-13) times 83

1078 and Level 6

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This Level 6 puzzle might be just a little bit easier than usual, so if you’ve never done this level before, be sure to try this one! You can succeed if you stick with it!

Print the puzzles or type the solution in this excel file: 10-factors-1073-1079

Here are a few facts about the number 1078:

  • 1078 is a composite number.
  • Prime factorization: 1078 = 2 × 7 × 7 × 11, which can be written 1078 = 2 × 7² × 11
  • 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 1078 has exactly 12 factors.
  • Factors of 1078: 1, 2, 7, 11, 14, 22, 49, 77, 98, 154, 539, 1078
  • Factor pairs: 1078 = 1 × 1078, 2 × 539, 7 × 154, 11 × 98, 14 × 77, or 22 × 49,
  • Taking the factor pair with the largest square number factor, we get √1078 = (√49)(√22) = 7√22 ≈ 32.83291

1 – 0 + 7 – 8 = 0 so 1078 is divisible by 11.

Since 11 is its largest prime factor we can make a lovely factor cake for 1078:

 

1078 is palindrome 4554 in BASE 6 because 4(6³) + 5(6²) + 5(6) + 4(1) = 1078

1077 and Level 5

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If you aren’t careful I might trick you into writing the numbers from 1 to 10 in the wrong places on this level 5 puzzle. Don’t let me trick you! Only write a factor if you know for sure where it goes. Study all the clues until logic directs you where to start.

Print the puzzles or type the solution in this excel file: 10-factors-1073-1079

Here is a little information about the number 1077:

It was difficult finding something unique about 1077, so I’m writing about a few things that I don’t usually mention:

1077 can be written as the difference of two squares two different ways:
539² – 538² = 1077
181² – 178² = 1077

1077 is the sum of two consecutive numbers 538 + 539 = 1077
1077 is the sum of three consecutive numbers 358 + 359 + 360
1077 is also the sum of three consecutive odd numbers 357-359-361

1077 is a leg in four Pythagorean triples:
1077-1436-1795 which is (3-4-5) times 359,
1077-193320-193323 which is 3 times (359-64440-64441),
1077-64436-64445 calculated from 181² – 178², 2(181)(178), 181² + 178², and
1077-579964-579965 calculated from 539² – 538², 2(539)(538), 539² + 538²

1076 and Level 4

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If you roll a pair of dice, you are taking a chance that the roll might not be favorable for you. Don’t take your chances when solving this puzzle. It can be solved completely by relying on logic. It won’t require any luck, but “Good luck!” anyway.

Print the puzzles or type the solution in this excel file: 10-factors-1073-1079

Here is some information about the number 1076:

  • 1076 is a composite number.
  • Prime factorization: 1076 = 2 × 2 × 269, which can be written 1076 = 2² × 269
  • 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 1076 has exactly 6 factors.
  • Factors of 1076: 1, 2, 4, 269, 538, 1076
  • Factor pairs: 1076 = 1 × 1076, 2 × 538, or 4 × 269
  • Taking the factor pair with the largest square number factor, we get √1076 = (√4)(√269) = 2√269 ≈ 32.80244

1076 is the hypotenuse of a Pythagorean triple:
276-1040-1076 which is 4 times (69-260-269)

1076 is a palindrome in a couple of bases:
It’s 434 in BASE 16 because 4(16²) + 3(16) +4(1) = 1076, and
it’s 1I1 in BASE 25 (I is 18 base 10) because 25² + 18(25) + 1 = 1076

1075 and Level 3

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Begin at the top of this level 3 puzzle and find the factors clue by clue until you reach the bottom. You can solve this puzzle and have fun doing it!

Print the puzzles or type the solution in this excel file: 10-factors-1073-1079

Here are some facts about the number 1075:

  • 1075 is a composite number.
  • Prime factorization: 1075 = 5 × 5 × 43, which can be written 1075 = 5² × 43
  • 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 1075 has exactly 6 factors.
  • Factors of 1075: 1, 5, 25, 43, 215, 1075
  • Factor pairs: 1075 = 1 × 1075, 5 × 215, or 25 × 43
  • Taking the factor pair with the largest square number factor, we get √1075 = (√25)(√43) = 5√43 ≈ 32.78719

If you had 43 quarters, you would have $10.75.
If you had 215 nickles, you would also have $10.75.

1075 is the hypotenuse of two Pythagorean triples:
645-860-1075 which is (3-4-5) times 215
301-1032-1075 which is (7-24-25) times 43

1075 is a palindrome in two consecutive bases:
It’s 898 in BASE 11 because 8(121) + 9(11) + 8(1) = 1075, and
it’s 757 in BASE 12 because 7(144) + 5(12) + 7(1) = 1075