There are two common factors of 10 and 14, but only one of them will put only numbers from 1 to 12 in the first column. Do you know what that factor is? If you do, figure out where to put the factors of 22, 66, 15 and so forth to make this puzzle function like a multiplication table. Each number from 1 to 12 can only appear once in the first column and once in the top row. You can do this!

Print the puzzles or type the solution in this excel file: 12 factors 1134-1147

Now I’d like to share some facts about the number 1138:

• 1138 is a composite number.
• Prime factorization: 1138 = 2 × 569
• 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 1138 has exactly 4 factors.
• Factors of 1138: 1, 2, 569, 1138
• Factor pairs: 1138 = 1 × 1138 or 2 × 569
• 1138 has no square factors that allow its square root to be simplified. √1138 ≈ 33.73426

33² + 7² = 1138

1138 is the hypotenuse of a Pythagorean triple:
462-1040-1138 calculated from 2(33)(7), 33² – 7², 33² + 7²

The common factors of 54 and 60 are 1, 2, 3, and 6. Just one of those common factors will put only numbers from 1 to 10 in the top row. That’s the factor you need to choose. To complete the puzzle, all the numbers from 1 to 10 must go in both the first column and the top row. Can you solve this puzzle?

Print the puzzles or type the solution in this excel file: 10-factors-1121-1133

Here are a few facts about the number 1124:

• 1124 is a composite number.
• Prime factorization: 1124 = 2 × 2 × 281, which can be written 1124 = 2² × 281
• 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 1124 has exactly 6 factors.
• Factors of 1124: 1, 2, 4, 281, 562, 1124
• Factor pairs: 1124 = 1 × 1124, 2 × 562, or 4 × 281
• Taking the factor pair with the largest square number factor, we get √1124 = (√4)(√281) = 2√281 ≈ 33.52611

1124 is the hypotenuse of a Pythagorean triple:
640-924-1124 which is 4 times 160-231-281

If I asked you to tell me what is significant about this set of numbers {13, 16, 19, 22}, what would you say?

Perhaps you would tell me they make an arithmetic sequence in which the common difference is 3.

What you probably wouldn’t tell me is that 1124 is a palindrome in those four bases!
It’s 686 in BASE 13 because 6(13²) + 8(13) + 6(1) = 1124,
464 in BASE 16 because 4(16²) + 6(16) + 4(1) = 1124
323 in BASE 19 because 3(19²) + 2(19) + 3(1) = 1124, and
272 in BASE 22 because 2(22²) + 7(22) + 2(1) = 1124

You will have to know the 11 and 12 times tables to solve this Level 3 Find the Factors 1-12 puzzle, but I’m sure you can do it! Stick with it, and don’t give up! Start with the clues at the top of the puzzle and work down row by row until it’s completed.

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

1115 is not a clue just the puzzle number. Here are some facts about the number 1115:

• 1115 is a composite number.
• Prime factorization: 1115 = 5 × 223
• 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 1115 has exactly 4 factors.
• Factors of 1115: 1, 5, 223, 1115
• Factor pairs: 1115 = 1 × 1115 or 5 × 223
• 1115 has no square factors that allow its square root to be simplified. √1115 ≈ 33.39162

1115 is the sum of nine consecutive prime numbers:
103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 = 1115

1115 is the hypotenuse of a Pythagorean triple:
669-892-1115 which is (3-4-5) times 223

1115 is palindrome 2B2 in BASE 21 (B is 11 base 10)
because 2(21²) + 11(21) + 2(1) = 1115

If this were a Find the Factors 1-12 puzzle, the possible common factors for 12 and 48 would be 4, 6, and 12. But we can only have factors from 1 to 10 so only one of those common factors will work with this puzzle. If you know which one, you are well on your way to solving it.

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

Here are some facts about the number 1104:

• 1104 is a composite number.
• Prime factorization: 1104 = 2 × 2 × 2 × 2 × 3 × 23, which can be written 1104 = 2⁴ × 3 × 23
• The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1104 has exactly 20 factors.
• Factors of 1104: 1, 2, 3, 4, 6, 8, 12, 16, 23, 24, 46, 48, 69, 92, 138, 184, 276, 368, 552, 1104
• Factor pairs: 1104 = 1 × 1104, 2 × 552, 3 × 368, 4 × 276, 6 × 184, 8 × 138, 12 × 92, 16 × 69, 23 × 48, or 24 × 46
• Taking the factor pair with the largest square number factor, we get √1104 = (√16)(√69) = 4√69 ≈ 33.2265.

1104 is the sum of the sixteen prime numbers from 37 to 103. Do you know what those prime numbers are?

1104 is also the sum of eight consecutive primes and two consecutive primes:
113 + 127 + 131 + 137 + 139 + 149 +151 + 157  = 1104
547 + 557 = 1104

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

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

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