# 1142 and Level 5

If you start in the right place, you will be able to solve this puzzle using logic and basic multiplication facts. Don’t get discouraged, if you make a mistake. I know you can figure this one out! Print the puzzles or type the solution in this excel file: 12 factors 1134-1147

Here are some facts about the number 1142:

• 1142 is a composite number.
• Prime factorization: 1142 = 2 × 571
• 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 1142 has exactly 4 factors.
• Factors of 1142: 1, 2, 571, 1142
• Factor pairs: 1142 = 1 × 1142 or 2 × 571
• 1142 has no square factors that allow its square root to be simplified. √1142 ≈ 33.79349 1142 is a palindrome in three different bases:
It’s 949 in BASE 11 because 9(11²) + 4(11) + 9(1) = 1142,
3G3 in BASE 17 (G is 16 base 10) because 3(17²) + 16(17) + 3(1) = 1142,
and 2H2 in BASE 20 (H is 17 base 10) because 2(20²) + 17(20) + 2(1) = 1142

# 1127 and Level 5

If the clues in this puzzle were in a Find the Factors 1 – 12, puzzle, the needed factors might be completely different than the ones in this puzzle’s solution. Fortunately, we can only use factors from 1 to 10, so this puzzle will make you think, but shouldn’t be so difficult. Print the puzzles or type the solution in this excel file: 10-factors-1121-1133

Here are a few facts about the number 1127:

• 1127 is a composite number.
• Prime factorization: 1127 = 7 × 7 × 23, which can be written 1127 = 7² × 23
• 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 1127 has exactly 6 factors.
• Factors of 1127: 1, 7, 23, 49, 161, 1127
• Factor pairs: 1127 = 1 × 1127, 7 × 161, or 23 × 49
• Taking the factor pair with the largest square number factor, we get √1127 = (√49)(√23) = 7√23 ≈ 33.57082 1127 is palindrome 5115 in BASE 6 because 5(6³) + 1(6²) + 1(6) + 5(1) = 1127

# 1117 and Level 5

Can you solve this Level 5 puzzle or will it make you feel like toast? Seriously, I think it’s a little easier than the previous puzzle, so give it a try. You might just become the toast of the town! Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

Now I’ll share some information about the number 1117:

• 1117 is a prime number.
• Prime factorization: 1117 is prime.
• The exponent of prime number 1117 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1117 has exactly 2 factors.
• Factors of 1117: 1, 1117
• Factor pairs: 1117 = 1 × 1117
• 1117 has no square factors that allow its square root to be simplified. √1117 ≈ 33.42155

How do we know that 1117 is a prime number? If 1117 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1117 ≈ 33.4. Since 1117 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1117 is a prime number. 26² + 21² = 1117

1117 is the hypotenuse of a Pythagorean triple:
235-1092-1117 calculated from 26² – 21², 2(26)(21), 26² + 21²

Here’s another way we know that 1117 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 26² + 21² = 1117 with 26 and 21 having no common prime factors, 1117 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1117 ≈ 33.4. Since 1117 is not divisible by 5, 13, 17, or 29, we know that 1117 is a prime number.

1117 is palindrome 151 in BASE 31 because 1(31²) + 5(31) + 1(1) = 1117

# 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)

# 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.