1113 and Level 2

If you’ve never solved a Find the Factors puzzle before, this level 2 puzzle will be a good one to try.  Just make sure each number 1 to 12 is written in the top row and the first column and that those numbers and the clues in the puzzle form a multiplication table. You can fill in the rest of the table later or not at all. Have fun!

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

Here is some information about the number 1113:

  • 1113 is a composite number.
  • Prime factorization: 1113 = 3 × 7 × 53
  • 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 1113 has exactly 8 factors.
  • Factors of 1113: 1, 3, 7, 21, 53, 159, 371, 1113
  • Factor pairs: 1113 = 1 × 1113, 3 × 371, 7 × 159, or 21 × 53
  • 1113 has no square factors that allow its square root to be simplified. √1113 ≈ 33.36165

1113 is the hypotenuse of a Pythagorean triple:
588-945-1113 which is 21 times (28-45-53)

1113 is made with three consecutive digits in these two consecutive bases:
It’s 789 in BASE 12 because 7(144) + 8(12) + 9(1) = 1113, and
it’s 678 in BASE 13 because 6(169) + 7(13) + 8(1) = 1113

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1103 and Level 2

The fourteen clues you see in this puzzle are all you need to find all the factors from 1 to 10 and complete the multiplication table. Can you find all those factors?

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

Here are some facts about the number 1103:

  • 1103 is a prime number.
  • Prime factorization: 1103 is prime.
  • The exponent of prime number 1103 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1103 has exactly 2 factors.
  • Factors of 1103: 1, 1103
  • Factor pairs: 1103 = 1 × 1103
  • 1103 has no square factors that allow its square root to be simplified. √1103 ≈ 33.21144

How do we know that 1103 is a prime number? If 1103 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1103 ≈ 33.2. Since 1103 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1103 is a prime number.

1103 is the sum of the nineteen prime numbers from 19 to 101.

1103 is palindrome 191 in BASE 29 because 1(29²) + 9(29) + 1(1) = 1103

1096 and Level 2

There are 17 clues in this level 2 puzzle. Two of those clues are 60 and three of them are 8. In a regular 12 × 12 multiplication table, both of those numbers appear 4 times each. The factors for this multiplication table puzzle won’t be in the usual places. Can you figure out where they need to go?

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

Here is a little bit about the number 1096:

  • 1096 is a composite number.
  • Prime factorization: 1096 = 2 × 2 × 2 × 137, which can be written 1096 = 2³ × 137
  • 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 1096 has exactly 8 factors.
  • Factors of 1096: 1, 2, 4, 8, 137, 274, 548, 1096
  • Factor pairs: 1096 = 1 × 1096, 2 × 548, 4 × 274, or 8 × 137
  • Taking the factor pair with the largest square number factor, we get √1096 = (√4)(√274) = 2√274 ≈ 33.10589

1096 is the hypotenuse of one Pythagorean triple:
704-840-1096 which is 8 times (88-105-137)

1088 and Level 2

This puzzle has three rows with three numbers in each and three columns with three numbers in each. Find the biggest number that is 10 or less that is a common factor of each set of three numbers, and you will be well on your way of solving the entire puzzle. Can you do it?

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

Now I’ll share some information about the number 1088:

  • 1088 is a composite number.
  • Prime factorization: 1088 = 2 × 2 × 2 × 2 × 2 × 2 × 17, which can be written 1088 = 2⁶ × 17
  • The exponents in the prime factorization are 6, and 1. Adding one to each and multiplying we get (6 + 1)(1 + 1) = 7 × 2 = 14. Therefore 1088 has exactly 14 factors.
  • Factors of 1088: 1, 2, 4, 8, 16, 17, 32, 34, 64, 68, 136, 272, 544, 1088
  • Factor pairs: 1088 = 1 × 1088, 2 × 544, 4 × 272, 8 × 136, 16 × 68, 17 × 64, or 32 × 34
  • Taking the factor pair with the largest square number factor, we get √1088 = (√64)(√17) = 8√17 ≈ 32.98485

Since 1088 = 32 × 34, we know the next number will be a square number.

1088 is the hypotenuse of a Pythagorean triple:
512-960-1088 which is (8-15-17) times 64

1088 is the sum of two consecutive prime numbers:
541 +547 = 1088

1088 looks interesting when written in some other bases:
It’s 3113 in BASE 7 because 3(7³) + 1(7²) + 1(7) +3(1) = 1088,
WW in BASE 33 (W is 32 base 10) Because 32(33) + 32(1) = 32(33 + 1) = 1088,
and it’s W0 in BASE 34 because 32(34) = 1088

1074 and Level 2

Jump up and cheer! You can solve this puzzle! Simply write all the numbers from 1 to 10 in both the first column and the top row so that those numbers are the factors of the given clues.

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

Now I’ll tell you something about the number 1074:

  • 1074 is a composite number.
  • Prime factorization: 1074 = 2 × 3 × 179
  • 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 1074 has exactly 8 factors.
  • Factors of 1074: 1, 2, 3, 6, 179, 358, 537, 1074
  • Factor pairs: 1074 = 1 × 1074, 2 × 537, 3 × 358, or 6 × 179
  • 1074 has no square factors that allow its square root to be simplified. √1074 ≈ 32.77194

1074 is the short leg in these Pythagorean triples:
1074-1432-1790 which is (3-4-5) times 358
1074-32032-32050, a primitive calculated from 2(179)(3), 179² – 3², 179² + 3²
1074-96120-96126 which is 6 times (179-16020-16021)
1074-288368-288370 calculated from 2(537)(1), 537² – 1², 537² + 1²

1074 is a palindrome when it is written in two different bases:
It”s 3C3 in BASE 17 (C is 12 base 10) because 3(17²) + 12(17) + 3(1) = 1074,
and 181 in BASE 29 because 29² + 8(29) + 1 = 1074

1064 and Level 2

If you did yesterday’s puzzle, then you will recognize four of the clues in today’s puzzle. They will give you a good start in finding all the rest of factors. See how well you do on this one!

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

Now I’ll tell you a little bit about the number 1064:

Its 0 is an even digit, and its last two digits, 64, can be evenly divided by 8, so 1064 is also divisible by 8.

  • 1064 is a composite number.
  • Prime factorization: 1064 = 2 × 2 × 2 × 7 × 19, which can be written 1064 = 2³ × 7 × 19
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1064 has exactly 16 factors.
  • Factors of 1064: 1, 2, 4, 7, 8, 14, 19, 28, 38, 56, 76, 133, 152, 266, 532, 1064
  • Factor pairs: 1064 = 1 × 1064, 2 × 532, 4 × 266, 7 × 152, 8 × 133, 14 × 76, 19 × 56, or 28 × 38
  • Taking the factor pair with the largest square number factor, we get √1064 = (√4)(√266) = 2√266 ≈ 32.61901

The difference in the numbers in one of its factor pairs, 28 × 38, is exactly ten, so we are exactly 25 away from the next perfect square number:
33² – 5² = 1089 – 25 = 1064

I like the way 1064 looks in a couple of other bases:
It’s 888 in BASE 11 because 8(11² + 11 + 1) = 8(133) = 1064, and
it’s 248 in BASE 22 because 2(22²) + 4(22) + 8(1) = 1064

1057 and Level 2

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

1045 and Level 2

This puzzle consists of six sets of three numbers. Find the common factor of each set of clues so that ALL of the factors involved are a number from 1 to 12, and you’ll solve this puzzle. Have fun!

Here are a few facts about the number 1045:

Obviously, 1045 can be evenly divided by 5, but since 1-0+4-5 = 0, we know that 1045 is also divisible by 11.

  • 1045 is a composite number.
  • Prime factorization: 1045 = 5 × 11 × 19
  • 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 1045 has exactly 8 factors.
  • Factors of 1045: 1, 5, 11, 19, 55, 95, 209, 1045
  • Factor pairs: 1045 = 1 × 1045, 5 × 209, 11 × 95, or 19 × 55
  • 1045 has no square factors that allow its square root to be simplified. √1045 ≈ 32.32646

1045 is the hypotenuse of a Pythagorean triple:
627-836-1045 which is (3-4-5) times 209.

1045 is also palindrome 171 in BASE 29 because 1(29²) + 7(29) + 1(1) = 1045

1039 and Level 2

The eleven clues in this puzzle are enough to figure out where to place the factors and then complete the entire multiplication table. Try it. I know you can solve it!

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

Now here’s a little about the number 1039:

  • 1039 is a prime number.
  • Prime factorization: 1039 is prime.
  • The exponent of prime number 1039 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1039 has exactly 2 factors.
  • Factors of 1039: 1, 1039
  • Factor pairs: 1039 = 1 × 1039
  • 1039 has no square factors that allow its square root to be simplified. √1039 ≈ 32.23352

How do we know that 1039 is a prime number? If 1039 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1039 ≈ 32.2. Since 1039 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1039 is a prime number.

1039 is a palindrome when it is written in two different bases:
It’s 727 in BASE 12 because 7(144) + 2(12) + 7(1) = 1039, and
494 in BASE 15 because 4(225) + 9(15) + 4(1) = 1039

1032 How Many Twelves Are in a 12×12 Times Table?

How many 12’s are in a standard 12×12 times table? There are five 12’s in the puzzle below. Six, if you count the 12 in the title. Is that too many, just the right number, or are there even more?

This is only a level 2 puzzle so it won’t be difficult to solve. . . unless I’ve put in too many 12’s!

Hmm. . .Try solving the puzzle, then fill in the rest of the multiplication table. Then you will know for sure how many 12’s SHOULD be in the table.

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

The number 1032 is divisible by 12. Here are a few more facts about that number:

1032 is made with a zero and three consecutive numbers so it is divisible by 3.

The last two digits of 1032 are 32 so 1032 can be evenly divided by 4.

Since 32 is divisible by 8 and preceded by an even zero in 1032, our number is also divisible by 8.

As you will soon see, 1032 is divisible by even more numbers than those listed above. Here are three of its factor trees:

1032 looks interesting in a couple of different bases:
It’s 4440 in BASE 6 because 4(6³ + 6² + 6¹) = 4(258) = 1032, and
it’s palindrome 3003 in BASE 7 because 3(7³ + 7⁰) = 3(344) = 1032.

  • 1032 is a composite number.
  • Prime factorization: 1032 = 2 × 2 × 2 × 3 × 43, which can be written 1032 = 2³ × 3 × 43
  • The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1032 has exactly 16 factors.
  • Factors of 1032: 1, 2, 3, 4, 6, 8, 12, 24, 43, 86, 129, 172, 258, 344, 516, 1032
  • Factor pairs: 1032 = 1 × 1032, 2 × 516, 3 × 344, 4 × 258, 6 × 172, 8 × 129, 12 × 86, or 24 × 43
  • Taking the factor pair with the largest square number factor, we get √1032 = (√4)(√258) = 2√258 ≈ 32.12476