1107 and Level 5

Some of this puzzle might be a little tricky, but you won’t allow it to trick you, right? Of course not!

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

Let me tell you something about the number 1107:

  • 1107 is a composite number.
  • Prime factorization: 1107 = 3 × 3 × 3 × 41, which can be written 1107 = 3³ × 41
  • The exponents in the prime factorization are 1 and 3. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1107 has exactly 8 factors.
  • Factors of 1107: 1, 3, 9, 27, 41, 123, 369, 1107
  • Factor pairs: 1107 = 1 × 1107, 3 × 369, 9 × 123, or 27 × 41
  • Taking the factor pair with the largest square number factor, we get √1107 = (√9)(√123) = 3√123 ≈ 33.27161

1107 is the hypotenuse of a Pythagorean triple:
243-1080-1107 which is 27 times (9-40-41)

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1099 and Level 5

The allowable common factors fro 8 and 24 are 2, 4, and 8. Which one of those should you choose? Find a different place to start the puzzle and you shouldn’t have to guess and check to see if you were right.

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

Here are some facts about the number 1099:

  • 1099 is a composite number.
  • Prime factorization: 1099 = 7 × 157
  • 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 1099 has exactly 4 factors.
  • Factors of 1099: 1, 7, 157, 1099
  • Factor pairs: 1099 = 1 × 1099 or 7 × 157
  • 1099 has no square factors that allow its square root to be simplified. √1099 ≈ 33.15117

1099 = 1 + 0 + 999 + 99. Thank you Stetson.edu for that fun fact.

1099 is the sum of the 13 prime numbers from 59 to 109.
That’s a fact that would take little effort to memorize!

1099 is also the sum of these prime numbers:
139 + 149 +  151 + 157 + 163 + 167 + 173 = 1099
359 + 367 + 373 = 1099

1099 is the hypotenuse of a Pythagorean triple:
595-924-1099 which is 7 times (85-132-157)

1099 is repdigit 777 in BASE 12 because 7(12² + 12 + 1) = 7(157) = 1099
1099 is palindrome 4D4 in BASE 15 (D is 13 in base 10)
because 4(15²) + 13(15) + 4(1) = 1099

1091 and Level 5

Can you figure out where to put all the numbers from 1 to 10 in both the first column and the top row so that those factors and the clues can become a multiplication table? Some of the clues might be a little tricky, but I’m sure you can figure them all out.

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

Here are a few facts about the number 1091:

  • 1091 is a prime number.
  • Prime factorization: 1091 is prime.
  • The exponent of prime number 1091 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1091 has exactly 2 factors.
  • Factors of 1091: 1, 1091
  • Factor pairs: 1091 = 1 × 1091
  • 1091 has no square factors that allow its square root to be simplified. √1091 ≈ 33.03029

How do we know that 1091 is a prime number? If 1091 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1091 ≈ 33. Since 1091 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1091 is a prime number.

1091 is the first prime number in the prime triplet (1091, 1093, 1097). It is also the middle number in the prime triplet (1087, 1091, 1093).

1091 looks interesting when it is written in some other bases:
It’s 13331 in BASE 5 because 1(5⁴) + 3(5³) + 3(5²) + 3(5) + 1(1) = 1091,
3D3 in BASE 17 (D is 13 base 10) because 3(17²) + 13(17) + 3(1) = 1091,
and it’s 123 in BASE 32 because 1(32²) + 2(32) + 3(1) = 1091

1085 and Level 5

Here’s another tricky level 5 puzzle for you to solve. Use logic, not guess and check, and you’ll do great!

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

Here are a few facts about the number 1085:

  • 1085 is a composite number.
  • Prime factorization: 1085 = 5 × 7 × 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 1085 has exactly 8 factors.
  • Factors of 1085: 1, 5, 7, 31, 35, 155, 217, 1085
  • Factor pairs: 1085 = 1 × 1085, 5 × 217, 7 × 155, or 31 × 35
  • 1085 has no square factors that allow its square root to be simplified. √1085 ≈ 32.93934

31 × 35 = 1085 means we are  only 4 numbers away from the next perfect square and
33² – 2² = 1085

1085 is the hypotenuse of a Pythagorean triple:
651-868-1085 which is (3-4-5) times 217

1085 looks interesting when it is written using a different base:
It’s 5005 in BASE 6 because 5(6³ + 1) = 1085,
765 in BASE 12 because 7(144) + 6(12) + 5(1) = 1085,
656 in BASE 13 because 6(13²) + 5(13) + 6(1) = 1085
VV in BASE 34 (V is 31 base 10) because 31(34) + 31(1) = 1085, and
it’s V0 in BASE 35 because 31(35) = 1085

1077 and Level 5

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

1067 and Level 5

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

1061 and Level 5

Study the clues in the puzzle below. If you begin with the right set of clues, the puzzle can be solved quite easily, but if you don’t, you might get tripped up. Good luck!

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

Here are a few facts about the number 1061:

  • 1061 is a prime number. 1061 and 1063 are twin primes.
  • Prime factorization: 1061 is prime.
  • The exponent of prime number 1061 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1061 has exactly 2 factors.
  • Factors of 1061: 1, 1061
  • Factor pairs: 1061 = 1 × 1061
  • 1061 has no square factors that allow its square root to be simplified. √1061 ≈ 32.57299

How do we know that 1061 is a prime number? If 1061 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1061 ≈ 32.6. Since 1061 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1061 is a prime number.

1061 is the sum of the 17 prime numbers from 29 to 101, and it also is the sum of these three consecutive prime numbers: 349 + 353 + 359 = 1061

31² + 10² = 1061 so 1061 is the hypotenuse of a Pythagorean triple:
620-861-1061, a primitive calculated from 2(31)(10), 31² – 10², 31² + 10²

Here’s another way we know that 1061 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 10² = 1061 with 31 and 10 having no common prime factors, 1061 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1061 ≈ 32.6. Since 1061 is not divisible by 5, 13, 17, or 29, we know that 1061 is a prime number.

 

 

 

 

1048 and Level 5

What is the common factor needed for 8 and 24 to make this puzzle work? Is it 2, 4, or 8? How about for 12 and 36? Is it 3, 4, 6, or 12? Don’t guess which common factor to use. In each case, all but one of the choices will be eliminated by using logic. It won’t be easy, but if you are determined, you can solve this puzzle.

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

Here are some facts about the number 1048:

  • 1048 is a composite number.
  • Prime factorization: 1048 = 2 × 2 × 2 × 131, which can be written 1048 = 2³ × 131
  • The exponents in the prime factorization are 1 and 3. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1048 has exactly 8 factors.
  • Factors of 1048: 1, 2, 4, 8, 131, 262, 524, 1048
  • Factor pairs: 1048 = 1 × 1048, 2 × 524, 4 × 262, or 8 × 131
  • Taking the factor pair with the largest square number factor, we get √1048 = (√4)(√262) = 2√262 ≈ 32.372828

1048 can be written as the difference of two squares two different ways:
263² – 261² = 1048
133² – 129² = 1048

1048 can also be expressed as 2 times a factor pair 3 different ways:
2(524)(1)
2(262)(2)
2(131)(4)

Those facts make 1048 a leg in these FIVE obscure Pythagorean triples:
1048-137286-137290 calculated from 263² – 261², 2(263)(261), 263² + 261²
1048-34314-34330 calculated from 133² – 129², 2(133)(129), 133² + 129²
1048-274575-274577 calculated from 2(524)(1), 524² – 1², 524² + 1²
1048-68640-68648 calculated from 2(262)(2), 262² – 2², 262² + 2²
1048-17145-17177 calculated from 2(131)(4), 131² – 4², 131² + 4²

1038 Hoppy Easter/April Fool’s Day

A trickster Easter bunny left some of my grandchildren candy and other treasures, not in a traditional Easter basket but in pink, green, and orange pumpkins! It’s April Fool’s Day so we shouldn’t be surprised.

That same trickster bunny has a purple pumpkin puzzle for YOU to try today, too. Part of the puzzle is easy while other parts are tricky: Is it 3 or 5 that is the common factor of 30 and 15 that makes this puzzle work?

Hmm…can you figure out where to put the numbers 1 to 10 in the first column and the top row or will you be tricked this April Fool’s Day?

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

Now I’ll tell you a little bit about the number 1038.

1 + 8 is divisible by 3 so 1038 is also divisible by 3. (Including multiples of 3 in the sum isn’t necessary for that divisibility trick to work.)

  • 1038 is a composite number.
  • Prime factorization: 1038 = 2 × 3 × 173
  • 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 1038 has exactly 8 factors.
  • Factors of 1038: 1, 2, 3, 6, 173, 346, 519, 1038
  • Factor pairs: 1038 = 1 × 1038, 2 × 519, 3 × 346, or 6 × 173
  • 1038 has no square factors that allow its square root to be simplified. √1038 ≈ 32.218007

1038 is also the hypotenuse of a Pythagorean triple:
312-990-1038 which is 6 times (52-165-173)

1036 Look, Look to the Rainbow

Finian’s Rainbow is a wonderful movie to enjoy on Saint Patrick’s Day. One of its songs reminds us to “Look, look to the rainbow”.

If you look to this rainbow, you will find all the factors of 1036:

There is a simple symmetry in every rainbow. There is also symmetry in palindromes which are numbers, words, or sentences that read the same forward or backward.

1036 demonstrates that symmetry when it is written in some other bases:
It’s repdigit 4444 in BASE 6 because 4(6³ + 6² + 6¹ + 6⁰) = 4(216 + 36 + 6 + 1) = 4(259) = 1036,
It’s 232 in BASE 22 because 2(22²) + 3(22) + 2(1) = 1036,
1M1 in BASE 23 (M is 22 base 10) because 23² + 22(23) + 1 = 1036, and
SS in BASE 36 (S is 28 base 10) because 28(36 + 1) = 28(37) = 1036

1036 is also the hypotenuse of a Pythagorean triple:
336-980-1036 which is 28 times 12-35-37

  • 1036 is a composite number.
  • Prime factorization: 1036 = 2 × 2 × 7 × 37, which can be written 1036 = 2² × 7 × 37
  • 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 1036 has exactly 12 factors.
  • Factors of 1036: 1, 2, 4, 7, 14, 28, 37, 74, 148, 259, 518, 1036
  • Factor pairs: 1036 = 1 × 1036, 2 × 518, 4 × 259, 7 × 148, 14 × 74, or 28 × 37,
  • Taking the factor pair with the largest square number factor, we get √1036 = (√4)(√259) = 2√259 ≈ 32.18695

There wasn’t a pot of gold at the end of our factor rainbow, but there is one here at the end of this post. It’s a level 5 puzzle, but it isn’t too difficult, so see if you can find all the factors that make the puzzle function like a multiplication table.

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