1108 and Level 6

This level 6 puzzle has eleven clues. Which ones give away the most logical place for you to start it? I hope you have a lot of fun solving this one!

Print the puzzles or type the solution in this excel file: 10-factors-1102-1110

Now I’ll share some facts about the number 1108:

  • 1108 is a composite number.
  • Prime factorization: 1108 = 2 × 2 × 277, which can be written 1108 = 2² × 277
  • 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 1108 has exactly 6 factors.
  • Factors of 1108: 1, 2, 4, 277, 554, 1108
  • Factor pairs: 1108 = 1 × 1108, 2 × 554, or 4 × 277
  • Taking the factor pair with the largest square number factor, we get √1108 = (√4)(√277) = 2√277 ≈ 33.28663

1108 is the hypotenuse of a Pythagorean triple:
460-1008-1108 which is 4 times (115-252-277)

1108 is a palindrome when it is written in three other bases:
It’s 454 in BASE 16 because 4(16²) + 5(16) + 4(1) = 1108,
3E3 in BASE 17 (E is 14 base 10) because 3(17²) +14(17) +3(1) = 1108, and
1E1 in BASE 27 because 27² + 14(27) + 1 = 1108

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1101 and Level 6

Which common factor of 6 and 24 will help you solve this puzzle? 2, 3, or 6?
Likewise, possible common factors of 8 and 4 are 1, 2, and 4, and for 48 and 36, you must choose between 4, 6, and 12.

In each case, only one choice will work with all the other clues in the puzzle.  You can figure out the correct choices and complete the entire puzzle by using logic. Good luck!

Print the puzzles or type the solution in this excel file: 12 factors 1095-1101

Here are a few facts about the number 1101:

Since it is made with a zero and exactly three identical numbers, it is divisible by 3.

  • 1101 is a composite number.
  • Prime factorization: 1101 = 3 × 367
  • 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 1101 has exactly 4 factors.
  • Factors of 1101: 1, 3, 367, 1101
  • Factor pairs: 1101 = 1 × 1101 or 3 × 367
  • 1101 has no square factors that allow its square root to be simplified. √1101 ≈ 33.18132

1101 is a palindrome when it is written in two other bases:
It’s 373 in BASE 18 because 3(18²) + 7(18) + 3(1) = 1101,
and 1J1 in BASE 25 (J is 19 in base 10) because 25² + 19(25) + 1 = 1101

1094 and Level 6

Should you choose 4 or 8 as the common factor of 32 and 16 in this puzzle?
Is 3 or 9 the common factor needed for 9 and 18?
And is 4 or 6 the common factor for 36 and 12 that will make this puzzle work?
In each of those cases, only one of those factors will work. Which one will it be?

The other clues will help you know where to logically start this puzzle. There is no need to guess and check. The entire puzzle can be solved using logic. Have fun!

Print the puzzles or type the solution in this excel file: 10-factors-1087-1094

Now I’ll tell you something about the number 1094:

  • 1094 is a composite number.
  • Prime factorization: 1094 = 2 × 547
  • 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 1094 has exactly 4 factors.
  • Factors of 1094: 1, 2, 547, 1094
  • Factor pairs: 1094 = 1 × 1094 or 2 × 547
  • 1094 has no square factors that allow its square root to be simplified. √1094 ≈ 33.07567

1094 is palindrome 2A2 in BASE 21 (A is 10 base 10) because 2(21²) + 10(21) + 2(1) = 1094

 

1086 and Level 6

The Find the Factors 1-12 puzzles allow 36 and 72 to use three common factors: 6, 9, and 12. Which one should you choose?  Likewise, 16 and 4’s permitted common factors are 2 and 4.

Sure you could guess and check to solve this puzzle, but that might frustrate you. I assure you that you can solve this puzzle by using logic.

If you get stuck getting started, here are a couple of things to consider:
Can both 72’s in the puzzle be 8×9? How about 6×12?
(36 = 3 × 12), (30 = 3 × 10), and (33 = 3 × 11) Which two of those clues MUST use both of the 3’s?

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

Now I’ll share a few facts about the number 1086:

  • 1086 is a composite number.
  • Prime factorization: 1086 = 2 × 3 × 181
  • 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 1086 has exactly 8 factors.
  • Factors of 1086: 1, 2, 3, 6, 181, 362, 543, 1086
  • Factor pairs: 1086 = 1 × 1086, 2 × 543, 3 × 362, or 6 × 181
  • 1086 has no square factors that allow its square root to be simplified. √1086 ≈ 32.95451

1086 is the hypotenuse of a Pythagorean triple:
114-1080-1086 which is 6 times (19-180-181)

1086 is a palindrome in three other bases:
It’s 8A8 in BASE 11 (A is 10 base 10) because 8(121) + 10(11) + 8(1) = 1086,
303 in BASE 19 because 3(19² + 1) = 1086, and
141 in BASE 31 because 1(31²) + 4(31) + 1(1) = 1086

1078 and Level 6

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

1068 and Level 6

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 × 5 × 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)

1062 Complicated Logic

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

1049 and Level 6

Find the Factors Puzzles are always solved using logic. Can you see the logic needed to solve this one?

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

Here are a few facts about the number 1049:

  • 1049 is a prime number.
  • Prime factorization: 1049 is prime.
  • The exponent of prime number 1049 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1049 has exactly 2 factors.
  • Factors of 1049: 1, 1049
  • Factor pairs: 1049 = 1 × 1049
  • 1049 has no square factors that allow its square root to be simplified. √1049 ≈ 32.38827

How do we know that 1049 is a prime number? If 1049 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1049 ≈ 32.4. Since 1049 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1049 is a prime number.

1049 is also the sum of three consecutive prime numbers:
347 + 349 + 353 = 1049

32² + 5² = 1049 so 1049 is the hypotenuse of a Pythagorean triple:
320-999-1049 calculated from 2(32)(5), 32² – 5², 32² + 5²

Here’s another way we know that 1049 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 32² + 5² = 1049 with 32 and 5 having no common prime factors, 1049 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1049 ≈ 32.4. Since 1049 is not divisible by 5, 13, 17, or 29, we know that 1049 is a prime number.

 

 

1042 and Level 6

I’ve already published the two level 5 puzzles that are in this week’s set of puzzles. If going from a level 4 puzzle to a level 6 puzzle is too big of a jump for you, then try either one of those two level 5 puzzles first. You can find them as well as this puzzle in the link below the puzzle.

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

Here are a few facts about the number 1042:

  • 1042 is a composite number.
  • Prime factorization: 1042 = 2 × 521
  • 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 1042 has exactly 4 factors.
  • Factors of 1042: 1, 2, 521, 1042
  • Factor pairs: 1042 = 1 × 1042 or 2 × 521
  • 1042 has no square factors that allow its square root to be simplified. √1042 ≈ 32.28002

31²  + 9² = 1042

1042 is the hypotenuse of a Pythagorean triple:
558-880-1042 calculated from 2(31)(9), 31²  – 9², 31²  + 9²

1042 is also a palindrome in a couple of bases:
It’s 868 in BASE 11 because 8(121) + 6(11) + 8(1) = 1042, and
2C2 in BASE 20 (C is 12 in base 10) because 2(400) + 10(20) + 2(1) = 1042

 

 

 

1018 and Level 6

Level 6 puzzles can be tricky to solve, but I promise that you can still solve this one using logic and knowledge of the basic multiplication table. Just write the numbers from 1 to 12 in both the first column and the top row so that the puzzle is like a partially filled out multiplication table with the factors in a different order. Like always, there is only one solution. Can you find it?

Print the puzzles or type the solution in this excel file: 12 factors 1012-1018

Look at these interesting facts about the number 1018:

27² + 17² = 1018
That means that 1018 is the hypotenuse of a Pythagorean triple:
440-918-1018 calculated from 27² – 17², 2(27)(17), 27² + 17²

It also means that (44² – 10²)/2 = 1018
Note that 27 + 17 = 44 and 27 – 17 = 10

1018 is full house 33322 in BASE 4 because 3(4⁴) + 3(4³) + 3(4²) + 2(4¹) + 2(4⁰) = 3(256 + 64 + 16) + 2(4 + 1) = 1018

Since 1018 is the sum of odd squares, it is divisible by 2. Since those odd squares have no common prime factors, you only have to check to see if 1018 is divisible by any Pythagorean triple hypotenuses less than or equal to (√1018)/2 ≈ 15.953. It is not divisible by 5 or 13, therefore 1018 only has two prime factors: 2 and 1018/2.

  • 1018 is a composite number.
  • Prime factorization: 1018 = 2 × 509
  • 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 1018 has exactly 4 factors.
  • Factors of 1018: 1, 2, 509, 1018
  • Factor pairs: 1018 = 1 × 1018 or 2 × 509
  • 1018 has no square factors that allow its square root to be simplified. √1018 ≈ 31.90611