1121 and Level 1

If you’ve learned how to multiply and divide, then you can solve this puzzle. Just write the numbers from 1 to 10 in both the first column and the top row so that the clues and those factors make a multiplication table. You can definitely do this one!

Here are a few facts about the number 1121:

  • 1121 is a composite number.
  • Prime factorization: 1121 = 19 × 59
  • 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 1121 has exactly 4 factors.
  • Factors of 1121: 1, 19, 59, 1121
  • Factor pairs: 1121 = 1 × 1121 or 19 × 59
  • 1121 has no square factors that allow its square root to be simplified. √1121 ≈ 33.48134

1121 is a palindrome in two other bases:
It’s 1C1 in BASE 28 (C is 12 base 10) because 28² + 12(28) + 1 = 1121,
and it’s 131 in BASE 32 because 32² + 3(32) + 1 = 1121

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1111 and Level 1

This is puzzle number 1111, a number made with four 1’s. The puzzle number doesn’t usually have anything to do with the puzzle, but I made an exception this time:  One of the factors of 1111 is important in solving this particular level 1 puzzle. Have fun solving it!

Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

Here are a few things I’ve learned about the number 1111:

  • 1111 is a composite number.
  • Prime factorization: 1111 = 11 × 101
  • 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 1111 has exactly 4 factors.
  • Factors of 1111: 1, 11, 101, 1111
  • Factor pairs: 1111 = 1 × 1111 or 11 × 101
  • 1111 has no square factors that allow its square root to be simplified. √111133.3316666

1111 is the hypotenuse of exactly one Pythagorean triple:
220-1089-1111 which is 11 times (20-99-101)

1111² = 1234321, a very special palindrome!

1111 is a repdigit in base 10, and it is a palindrome in three consecutive bases plus one more:
It’s 787 in BASE 12 because 7(144) + 8(12) + 7(1) = 1111,
676 in BASE 13 because 6(169) + 7(13) + 6(1) = 1111,
595 in BASE 14 because 5(196) + 9(14) + 5(1) = 1111, and it’s
171 in BASE 30 because 1(900) + 7(30) + 1(1) = 1111

 

1102 and Level 1

Write each number from 1 to 10 in both the first column and the top row so that those numbers are the factors of the given clues. This one is not too difficult, so if you haven’t solved one of these puzzles before, give it a try!

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

Here is some information about the number 1102:

  • 1102 is a composite number.
  • Prime factorization: 1102 = 2 × 19 × 29
  • 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 1102 has exactly 8 factors.
  • Factors of 1102: 1, 2, 19, 29, 38, 58, 551, 1102
  • Factor pairs: 1102 = 1 × 1102, 2 × 551, 19 × 58, or 29 × 38
  • 1102 has no square factors that allow its square root to be simplified. √1102 ≈ 33.19639

1102 = 2(29)(19) so we know that 480-1102-1202 is a Pythagorean triple
calculated from 29²-19², 2(29)(19), 29²+19²

1102 is also the hypotenuse of a Pythagorean triple:
760-798-1102 which is (20-21-29) times 38.

1102 is a palindrome when it is written in a couple of other bases:
It’s 2F2 in BASE 20 (F is 15 base 10) because 2(20²) + 15(20) + 2(1) = 1102,
and it’s 262 in BASE 22 because 2(22²) + 6(22) + 2(1) = 1102.

1095 and Level 1

This awkward-looking puzzle is as simple as clockwork to solve. Put a 5 above the 55 in the top row and another 5 before the 30 in the first column. Then write what number the big hand is pointing to when it is 55 minutes after the hour and so forth until you have written the numbers from 1 to 12 in both the first column and the top row. (You will have to figure out what numbers go with the column and row without clues.)

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

That puzzle had something to do with a clock, while the number 1095 has something to do with the number of days in three non-leap years. Here are some other facts about it:

1 + 0 + 9 + 5 = 15, so 1095 can be evenly divided by 3. Since its last digit is 5, it is also divisible by 5.

  • 1095 is a composite number.
  • Prime factorization: 1095 = 3 × 5 × 73
  • 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 1095 has exactly 8 factors.
  • Factors of 1095: 1, 3, 5, 15, 73, 219, 365, 1095
  • Factor pairs: 1095 = 1 × 1095, 3 × 365, 5 × 219, or 15 × 73
  • 1095 has no square factors that allow its square root to be simplified. √1095 ≈ 33.09078

1095 is the hypotenuse of FOUR Pythagorean triples:
81-1092-1095 which is 3 times (27-364-365)
228-1071-1095 which is 3 times (76-357-365)
657-876-1095 which is (3-4-5) times 219
720-825-1095 which is 15 times (48-55-73)

1089 Perfect Squares

The nine clues in today’s puzzle are all perfect squares. They are all you need to find all the factors that can turn this puzzle into a multiplication table . . . but with the rows and columns not in the typical order:

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

1089 is also a perfect square.

The previous perfect square was 32² = (33 – 1)² = 33² + 1 – 2(33) = 1024
The next perfect square will be 34² = (33 + 1)² = 33² + 1 + 2(33) = 1156

Here’s a little more about the number 1089:

  • 1089 is a composite number.
  • Prime factorization: 1089 = 3 × 3 × 11 × 11, which can be written 1089 = 3²× 11²
  • The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 × 3 = 9. Therefore 1089 has exactly 9 factors.
  • Factors of 1089: 1, 3, 9, 11, 33, 99, 121, 363, 1089
  • Factor pairs: 1089 = 1 × 1089, 3 × 363, 9 × 121, 11 × 99, or 33 × 33
  • 1089 is a perfect square. √1089 = 33

1, 9, 121, and 1089 are all perfect square factors of 1089.

1089 can be 3 × 3 perfect squares arranged on an 11 × 11 perfect square grid:

1089 can also be 11 × 11 perfect squares arranged on a 3 × 3 perfect square grid:

Not only is 1089 the 33rd perfect square, but it is also the sum of the first 33 odd numbers. Note that the nth perfect square is also the sum of the first odd numbers:

I’m not attempting to make a picture of this nine-sided shape, but 1089 is the 18th nonagonal number because 18(7(18) – 5)/ 2 = 1089,
or written another way 7(18²)/2 – 5(18)/2 = 1089.

1089 is the sum of five consecutive prime numbers:
199 + 211 + 223 + 227 + 229 = 1089

Stetson.edu informs us that 9 × 1089 = 9801

1089 looks rather square when it is written in several other bases:
It’s 900 in BASE 11 because 9(11²) = 1089,
441 in BASE 16 because  4(16²) + 4(16) + 1(1) = 1089,
169 in BASE 30 because 1(30²) + 6(30) + 9(1) = 1089,
144 in BASE 31 because 1(31²) + 4(31) + 4(1) = 1089,
121 in BASE 32 because 1(32²) + 2(32) + 1(1) = 1089,
100 in BASE 33 because 1(33²) = 1089

1081 and Level 1

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

1073 and Level 1

You might not recognize it, but this puzzle is just a multiplication table. The factors in the table are all missing, and they aren’t in their usual places, but you have everything you need here to find the factors from 1 to 10 and then complete the entire multiplication table. It’s a level 1 puzzle so I am absolutely sure you can solve it!

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

Here is a little information about the number 1073:

  • 1073 is a composite number.
  • Prime factorization: 1073 = 29 × 37
  • 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 1073 has exactly 4 factors.
  • Factors of 1073: 1, 29, 37, 1073
  • Factor pairs: 1073 = 1 × 1073 or 29 × 37
  • 1073 has no square factors that allow its square root to be simplified. √1073 ≈ 32.75668

32² + 7² = 1073
28² + 17² = 1073

Because 29 × 37 = 1073, it is the hypotenuse of FOUR Pythagorean triples, two of which are primitives:
348-1015-1073 which is 29 times (12-35-37)
740-777-1073 which is (20-21-29) times 37
448-975-1073 calculated from 2(32)(7), 32² – 7², 32² + 7²
495-952-1073 calculated from 28² – 17², 2(28)(17), 28² + 17²

1073 looks interesting when it is written in some other bases:
It’s 4545 in BASE 6 because 4(6³) + 5(6²) + 4(6) + 5(1) = 1073,
292 in BASE 21 because 2(21²) + 9(21) + 2(1) = 1073, and
TT in BASE 36 (T is 29 base 10) because 29(36) + 29(1) = 29(37) = 1073

1063 and Level 1

Lucky you found this puzzle today! You can solve it by writing the factors 1 to 12 in both the first column and the top row so that the given clues are the products of the corresponding factors.

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

Now I’ll share a little information about the number 1063:

  • 1063 is a prime number. 1061 and 1063 are twin primes.
  • Prime factorization: 1063 is prime.
  • The exponent of prime number 1063 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1063 has exactly 2 factors.
  • Factors of 1063: 1, 1063
  • Factor pairs: 1063 = 1 × 1063
  • 1063 has no square factors that allow its square root to be simplified. √1063 ≈ 32.60368

How do we know that 1063 is a prime number? If 1063 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1063 ≈ 32.6. Since 1063 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1063 is a prime number.

1063 is the sum of seven consecutive prime numbers:
137 + 139 + 149 + 151 + 157 + 163 + 167 = 1063

1063 is a high flying palindrome in one other base:
It’s 747 in BASE 12 because 7(12²) + 4(12) + 7(1) = 1063

1055 and Level 1

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

1044 and Level 1

All of the clues in this puzzle have three common factors, but only one of those three factors won’t put a number greater than twelve in either the first column or the top row. Can you figure out what that common factor is as well as all the other factors that belong in this puzzle?

Now I’ll share some information about the number 1044:

  • 1044 is a composite number.
  • Prime factorization: 1044 = 2 × 2 × 3 × 3 × 29, which can be written 1044 = 2² × 3² × 29
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1044 has exactly 18 factors.
  • Factors of 1044: 1, 2, 3, 4, 6, 9, 12, 18, 29, 36, 58, 87, 116, 174, 261, 348, 522, 1044
  • Factor pairs: 1044 = 1 × 1044, 2 × 522, 3 × 348, 4 × 261, 6 × 174, 9 × 116, 12 × 87, 18 × 58 or 29 × 36
  • Taking the factor pair with the largest square number factor, we get √1044 = (√36)(√29) = 6√29 ≈ 32.31099

30² + 12² =1044

1044 is the hypotenuse of a Pythagorean triple:
720-756-1044 calculated from 2(30)(12), 30² – 12², 30² + 12².
It is also (20-21-29) times 36.

1044 is the sum of twin primes: 521 + 523 = 1044

1044 looks interesting a few other bases:
It’s 414 in BASE 16 because 4(16²) + 1(16) + 4(1) = 1044,
TT in BASE 35 (T is 29 base 10) because 29(35) + 29(1) = 29(35 + 1) = 29(36) = 1044, and T0 in BASE 36 because 29(36) = 1044