Can you figure out where to put the numbers from 1 to 10 in both the first column and the top row so that the lights on this Christmas tree work properly?

# Level 5 Puzzle

# 1316 Hard Candy

When I have a cold or a cough, I often have a piece of hard candy in my mouth. These red cinnamon candies are tasty, but they aren’t very good for soothing throats! Will this red hot cinnamon candy puzzle be too hard for you to solve? It may be a little bit of a challenge, but I’m sure you can solve it if you let logic be your guide the entire time.

Print the puzzles or type the solution in this excel file: 12 factors 1311-1319

# 1256 and Level 5

Use logic, not guess and check, to find where the numbers from 1 to 12 belong in both the first column and the top row so that the puzzle acts like a multiplication table. Can you do it, or will some of the clues trick you?

Print the puzzles or type the solution in this excel file: 12 factors 1251-1258

Now I’ll share some facts about the number 1256:

- 1256 is a composite number.
- Prime factorization: 1256 = 2 × 2 × 2 × 157, which can be written 1256 = 2³ × 157
- The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1256 has exactly 8 factors.
- Factors of 1256: 1, 2, 4, 8, 157, 314, 628, 1256
- Factor pairs: 1256 = 1 × 1256, 2 × 628, 4 × 314, or 8 × 157
- Taking the factor pair with the largest square number factor, we get √1256 = (√4)(√314) = 2√314 ≈ 35.44009

1256 is the sum of two squares:

34² + 10² = 1256

1256 is the hypotenuse of a Pythagorean triple:

680-1056-1256 which is **8** times (85-132-**157**) and

can also be calculated from 2(34)(10), 34² – 10², 34² + 10²

1256 is 888 in BASE 12 because 8(144 + 12 + 1) = 8(157) = 1256

# 1237 and Level 5

Level 5 puzzles always have at least one set of clues with more than one possible common factor. Still only one of those factors will actually work with all the rest of the clues. Can you use logic to find the factors needed to solve this puzzle?

Print the puzzles or type the solution in this excel file: 12 factors 1232-1241

Let’s look at some facts about the number 1237:

- 1237 is a prime number.
- Prime factorization: 1237 is prime.
- The exponent of prime number 1237 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1237 has exactly 2 factors.
- Factors of 1237: 1, 1237
- Factor pairs: 1237 = 1 × 1237
- 1237 has no square factors that allow its square root to be simplified. √1237 ≈ 35.17101

**How do we know that ****1237**** is a prime number?** If 1237 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1237 ≈ 35.1. Since 1237 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1237 is a prime number.

1237 is the sum of two squares:

34² + 9² = 1237

1237 is the hypotenuse of a Pythagorean triple:

612-1075-1237 calculated from 2(34)(9), 34² – 9², 34² + 9²

Here’s another way we know that 1237 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 34² + 9² = 1237 with 34 and 9 having no common prime factors, 1237 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1237 ≈ 35.1. Since 1237 is not divisible by 5, 13, 17, or 29, we know that 1237 is a prime number.

# 1228 and Level 5

This level 5 puzzle has a row and a column with the exact same two clues. That ISN’T a good place to start this puzzle! Nevertheless, you can solve it, if you use logic and your knowledge of a basic 10 × 10 multiplication table. There is only one solution. Good luck!

Print the puzzles or type the solution in this excel file: 10-factors-1221-1231

Now I’ll share some information about the number 1228:

- 1228 is a composite number.
- Prime factorization: 1228 = 2 × 2 × 307, which can be written 1228 = 2
**²**× 307 - 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 1228 has exactly 6 factors.
- Factors of 1228: 1, 2, 4, 307, 614, 1228
- Factor pairs: 1228 = 1 × 1228, 2 × 614, or 4 × 307
- Taking the factor pair with the largest square number factor, we get √1228 = (√4)(√307) = 2√307 ≈ 35.04283

1228 is repdigit 444 in BASE 17 because 4(17² + 17 + 1) = 4(307) = 1228

# 1215 and Level 5

Level 5 puzzles are a little trickier than level 4 puzzles, but you can still use logic to find the unique solution to today’s puzzle. Stick with it!

Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

Here is some information about the number 1215:

- 1215 is a composite number.
- Prime factorization: 1215 = 3 × 3 × 3 × 3 × 3 × 5, which can be written 1215 = 3⁵ × 5
- The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 1215 has exactly 12 factors.
- Factors of 1215: 1, 3, 5, 9, 15, 27, 45, 81, 135, 243, 405, 1215
- Factor pairs: 1215 = 1 × 1215, 3 × 405, 5 × 243, 9 × 135, 15 × 81, or 27 × 45
- Taking the factor pair with the largest square number factor, we get √1215 = (√81)(√15) = 9√15 ≈ 34.85685

1215 is the hypotenuse of a Pythagorean triple:

729-972-1215 which is (3-4-**5**) times **243**

# 1205 and Level 5

These Level 5 puzzles always have at least one set of clues whose common factor can only be one number. Find it, and you’ll be able to proceed using just logic and basic multiplication facts. Have fun with it!

Print the puzzles or type the solution in this excel file: 10-factors-1199-1210

Here are some facts about the number 1205:

- 1205 is a composite number.
- Prime factorization: 1205 = 5 × 241
- 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 1205 has exactly 4 factors.
- Factors of 1205: 1, 5, 241, 1205
- Factor pairs: 1205 = 1 × 1205 or 5 × 241
- 1205 has no square factors that allow its square root to be simplified. √1205 ≈ 34.7131

26² + 23² = 1205

34² + 7² = 1205

1205 is the hypotenuse of FOUR Pythagorean triples:

147-1196-1205 calculated from 26² – 23², 2(26)(23), 26² + 23²

476-1107-1205 calculated from 2(34)(7), 34² – 7², 34² + 7²

600-1045-1205 which is 5 times (120-209-241)

723-964-1205 which is (3-4-5) times 241

# 1195 You Can Find the Answer in This Book

The new school year is underway. Much may have been forgotten over the summer. If you don’t quite remember all the multiplication tables, this puzzle book can help you remember them AND help your brain grow. You might still find it a challenge, but that only makes it more fun!

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Now I’ll share a few facts about the number 1195:

- 1195 is a composite number.
- Prime factorization: 1195 = 5 × 239
- 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 1195 has exactly 4 factors.
- Factors of 1195: 1, 5, 239, 1195
- Factor pairs: 1195 = 1 × 1195 or 5 × 239
- 1195 has no square factors that allow its square root to be simplified. √1195 ≈ 34.56877

1195 is also the hypotenuse of a Pythagorean triple:

717-956-1195 which is (3-4-**5**) times **239**

# 1167 and Level 5

Will some of the tricky clues in this level 5 puzzle fool you? They won’t if you only write factors of which you are 100% sure. Always use logic. Never guess and check.

Print the puzzles or type the solution in this excel file: 12 factors 1161-1173

Now I’ll write a little bit about the number 1167:

- 1167 is a composite number.
- Prime factorization: 1167 = 3 × 389
- 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 1167 has exactly 4 factors.
- Factors of 1167: 1, 3, 389, 1167
- Factor pairs: 1167 = 1 × 1167 or 3 × 389
- 1167 has no square factors that allow its square root to be simplified. √1167 ≈ 34.16138

1167 is the hypotenuse of a Pythagorean triple:

567-1020-1167 which is **3** times (189-340-**389**)

1167 is palindrome 5D5 in BASE 14 (D is 13 base 10)

because 5(14²) + 13(14) + 5(1) = 1167

# 1154 and Level 5

I’m sure you can have a lot of fun solving this puzzle. Remember to use logic before you write down any of the factors, and it will be fun instead of frustrating.

Print the puzzles or type the solution in this excel file: 10-factors-1148-1160

Now I’ll share some facts about the number 1154:

- 1154 is a composite number.
- Prime factorization: 1154 = 2 × 577
- 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 1154 has exactly 4 factors.
- Factors of 1154: 1, 2, 577, 1154
- Factor pairs: 1154 = 1 × 1154 or 2 × 577
- 1154 has no square factors that allow its square root to be simplified. √1154 ≈ 33.97058

1154 is the sum of eight consecutive prime numbers:

127 + 131 + 137 + 139 + 149 + 151 + 157 + 163 = 1154

25² + 23² = 1154

1154 is the hypotenuse of a Pythagorean triple:

96-1150-1154 calculated from 25² – 23², 2(25)(23), 25² + 23²

1154 is palindrome 202 in BASE 24

because 2(24²) + 2(1) = 2(24² + 1) = 2(577) = 1154

Did you notice that 1154 has a relationship with 23², 24², and 25²?