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

This puzzle has both 54 and 56 as clues. Many people get the factors involved (6, 7, 8, 9) mixed up. Remember these two multiplication facts:
6 7 × 8 9 = 54
6 7 × 8 9 = 56
The closer factors make the bigger number.

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

Now I’ll share some information about the number 1090:

• 1090 is a composite number.
• Prime factorization: 1090 = 2 × 5 × 109
• 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 1090 has exactly 8 factors.
• Factors of 1090: 1, 2, 5, 10, 109, 218, 545, 1090
• Factor pairs: 1090 = 1 × 1090, 2 × 545, 5 × 218, or 10 × 109
• 1090 has no square factors that allow its square root to be simplified. √1090 ≈ 33.01515

1090 is the sum of the fourteen prime numbers from 47 to 107.

1090 is the sum of two squares two different ways:
27² + 19² = 1090
33² +  1² = 1090

1090 is the hypotenuse of four Pythagorean triples:
66-1088-1090 calculated from 2(33)(1), 33² –  1², 33² +  1²;
it is also 2 times (33-544-545),
368-1026-1090 calculated from 27² – 19² , 2(27)(19) , 27² + 19²;
it is also 2 times (184-513-545),
600-910-1090 which is 10 times (60-91-109)
654-872-1090 which is (3-4-5) times 218

1090 is a palindrome in two different bases:
It’s 1441 in BASE 9 because 1(9³) + 4(9²) + 4(9) + 1(1) = 1090
101 in BASE 33 because 33² + 1 = 1090

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

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

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

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

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