Many people fly home or away from home for the holidays. Here’s a puzzle to occupy some of your time while you’re in flight.

# Level 6 Puzzle

# 1320 Christmas Factor Trees

Regardless of its size, an evergreen tree is a mighty symbol at Christmastime. Today’s factoring puzzle features a couple of relatively small Christmas trees, but don’t think for even one minute that these little trees make for an easy puzzle. It’s a level 6 puzzle so there are several places that the clues might trick you. Use logic through the entire process, and you should be able to solve it!

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

That factoring puzzle has nothing to do with the factors of 1320. In case you are looking for factor trees for the number 1320, here are a few of the MANY possible ones:

# 1257 and Level 6

Both 6 and 12 are allowable common factors of 60 and 12. Likewise, both 8 and 12 are allowable common factors of 96 and 72. In each case, only one of those common factors will work with this puzzle. Don’t guess and check each one. Study the other clues and at least one wrong common factor will be eliminated. Have fun solving it!

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

Now I’ll write a few things about the number 1257:

- 1257 is a composite number.
- Prime factorization: 1257 = 3 × 419
- 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 1257 has exactly 4 factors.
- Factors of 1257: 1, 3, 419, 1257
- Factor pairs: 1257 = 1 × 1257 or 3 × 419
- 1257 has no square factors that allow its square root to be simplified. √1257 ≈ 35.4542

1257 is the difference of two squares two different ways:

211² – 208² = 1257

629² – 628² = 1257

1257 is palindrome 393 in BASE 19

# 1250 and Level 6

The clues in one of the columns for this puzzle as well as one of the rows are 9 and 3. You will need to figure out where to put the factors 1, 3, 3, and 9 to make those clues work. You might think it doesn’t matter where you write those factors, but believe me, it does matter. My advice: Don’t start with those clues. Find another logical place to start. Good luck with this one!

Print the puzzles or type the solution in this excel file: 10-factors-1242-1250

Here are some facts about the number 1250:

- 1250 is a composite number.
- Prime factorization: 1250 = 2 × 5 × 5 × 5 × 5, which can be written 1250 = 2 × 5⁴
- The exponents in the prime factorization are 1 and 5. Adding one to each and multiplying we get (1 + 1)(4 + 1) = 2 × 5 = 10. Therefore 1250 has exactly 10 factors.
- Factors of 1250: 1, 2, 5, 10, 25, 50, 125, 250, 625, 1250
- Factor pairs: 1250 = 1 × 1250, 2 × 625, 5 × 250, 10 × 125, or 25 × 50
- Taking the factor pair with the largest square number factor, we get √1250 = (√625)(√2) = 25√2 ≈
**35**.**3553**4

1250 is the sum of consecutive squares two different ways:

193 + 197 + 199 + 211 + 223 + 227 = 1250

619 + 631 = 1250

1250 is the sum of two squares THREE different ways:

31² + 17² = 1250

25² + 25² = 1250

35² + 5² = 1250

1250 is the hypotenuse of FOUR Pythagorean triples:

750-1000-1250 which is (3-4-**5**) times **250**,

672-1054-1250 which is **2** times (336-527-**625**) and

can also be calculated from 31² – 17², 2(31)(17), 31² + 17²,

440-1170-1250 which is **10** times (44-117-**125**), and

350-1200-1250 which is (7-24-**25**) times **50** and

can also be calculated from 2(35)(5), 35² – 5², 35² + 5²

# 1238 and Level 6

If you use logic, you can figure out the solution to this puzzle. You will have to study all the clues just to know where to start, but I think you’ll find a lot of satisfaction in finding the solution.

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

Now I’ll share some facts about the number 1238:

- 1238 is a composite number.
- Prime factorization: 1238 = 2 × 619
- 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 1238 has exactly 4 factors.
- Factors of 1238: 1, 2, 619, 1238
- Factor pairs: 1238 = 1 × 1238 or 2 × 619
- 1238 has no square factors that allow its square root to be simplified. √1238 ≈ 35.18522

Because of its prime factors, I know that 1238 is part of only one Pythagorean triple:

1238-383160-383162

1238 is a palindrome in two other bases:

It’s 646 in BASE 14, and

it’s 383 in BASE 19.

# 1229 and Level 6

The only common factors permitted for 32 and 40 in this puzzle are 4 and 8, but which one will work for this puzzle? Likewise, you must decide if 3 or 6 is the right common factor for 18 and 30. Don’t guess which factor to use. Study the other clues and let logic guide your decisions until the unique solution is found. Have fun with this one!

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

This is my 1229th post, so I will tell you a little bit about the number 1229:

- 1229 is a prime number.
- Prime factorization: 1229 is prime.
- The exponent of prime number 1229 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1229 has exactly 2 factors.
- Factors of 1229: 1, 1229
- Factor pairs: 1229 = 1 × 1229
- 1229 has no square factors that allow its square root to be simplified. √1229 ≈ 35.057

**How do we know that ****1229**** is a prime number?** If 1229 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1229 ≈ 35.1. Since 1229 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1229 is a prime number.

1229 is the sum of three consecutive prime numbers:

401 + 409 + 419 = 1229

1229 is the sum of two square numbers:

35² + 2² = 1229

1229 is the hypotenuse of a primitive Pythagorean triple:

140-1221-1229 calculated from 2(35)(2), 35² – 2², 35² + 2²

Here’s another way we know that 1229 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 35² + 2² = 1229 with 35 and 2 having no common prime factors, 1229 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1229 ≈ 35.1. Since 1229 is not divisible by 5, 13, 17, or 29, we know that 1229 is a prime number.

# 1216 and Level 6

Is 6 or 9 the needed common factor for 54 and 36?

The other one will be the common factor for 72 and 18.

Will 5 or 10 be the common factor of 30 and 50?

Will 2 or 4 be the common factor of 16 and 4?

Don’t guess the answers to those questions! The other clues in the puzzle will help you find the answers. Will you be stumped or will you triumph?

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

Now I’ll share some information about the number 1216:

- 1216 is a composite number.
- Prime factorization: 1216 = 2 × 2 × 2 × 2 × 2 × 2 × 19, which can be written 1216 = 2⁶ × 19
- 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 1216 has exactly 14 factors.
- Factors of 1216: 1, 2, 4, 8, 16, 19, 32, 38, 64, 76, 152, 304, 608, 1216
- Factor pairs: 1216 = 1 × 1216, 2 × 608, 4 × 304, 8 × 152, 16 × 76, 19 × 64, or 32 × 38
- Taking the factor pair with the largest square number factor, we get √1216 = (√64)(√19) = 8√19 ≈ 34.87119

1216 is the sum of the ten prime numbers between 100 and 150:

101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 = 1216

1216 is the 19th nonagonal number because

19(7·19 – 5)/2 = 1216, in other words because 19 × 64 = 1216

# 1206 and Level 6

If you carefully study all the clues in this Level 6 puzzle and use logic, you should be able to solve the puzzle. Stick with it and you’ll succeed!

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

Here are some facts about the number 1206:

- 1206 is a composite number.
- Prime factorization: 1206 = 2 × 3 × 3 × 67, which can be written 1206 = 2 × 3² × 67
- The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1206 has exactly 12 factors.
- Factors of 1206: 1, 2, 3, 6, 9, 18, 67, 134, 201, 402, 603, 1206
- Factor pairs: 1206 = 1 × 1206, 2 × 603, 3 × 402, 6 × 201, 9 × 134, or 18 × 67,
- Taking the factor pair with the largest square number factor, we get √1206 = (√9)(√134) = 3√134 ≈ 34.72751

Notice that 6·201 = 1206. Not very many numbers can equal themselves by using their own digits in a different way with +, -, ×, ÷, and/or parenthesis. That fact makes 1206 only the seventeenth Friedman Number.

1206 is also the sum of the twenty prime numbers from 19 to 103.

# 1196 and Level 6

In this puzzle, the permissible common factors of 48 and 72 are 6, 8, and 12. For clues 8 and 16, you can choose from common factors 2, 4, or 8. Which choices will make the puzzle work? I’m not telling, but I promise that the entire puzzle can be solved using logic and a basic knowledge of a 12×12 multiplication table. There is only one solution.

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

Here are some facts about the number 1196:

- 1196 is a composite number.
- Prime factorization: 1196 = 2 × 2 × 13 × 23, which can be written 1196 = 2² × 13 × 23
- The exponents in the prime factorization are 2, 1, and 1. Adding one to each and multiplying we get (2 + 1)(1 + 1)(1 + 1) = 3 × 2 × 2 = 12. Therefore 1196 has exactly 12 factors.
- Factors of 1196: 1, 2, 4, 13, 23, 26, 46, 52, 92, 299, 598, 1196
- Factor pairs: 1196 = 1 × 1196, 2 × 598, 4 × 299, 13 × 92, 23 × 52, or 26 × 46
- Taking the factor pair with the largest square number factor, we get √1196 = (√4)(√299) = 2√299 ≈ 34.58323

1196 is the hypotenuse of a Pythagorean triple:

460-1104-1196 which is (5-12-**13**) times **92**

1196 is a palindrome in three different bases:

It’s 14241 in BASE 5,

838 in BASE 12, and

616 in BASE 14

# 1182 A Unique Solution

Every Find the Factors puzzle I make has a unique solution. That fact is central to the logic needed to begin this particular puzzle. I hope it frustrates you a little, but not too much. Then when you finally solve it, it will be so much sweeter!

Print the puzzles or type the solution in this excel file: 10-factors-1174-1186

Here are some facts about the number 1182:

- 1182 is a composite number.
- Prime factorization: 1182 = 2 × 3 × 197
- 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 1182 has exactly 8 factors.
- Factors of 1182: 1, 2, 3, 6, 197, 394, 591, 1182
- Factor pairs: 1182 = 1 × 1182, 2 × 591, 3 × 394, or 6 × 197
- 1182 has no square factors that allow its square root to be simplified. √1182 ≈ 34.38023

1182 is the hypotenuse of a Pythagorean triple:

168-1170-1182 which is **6** times (28-195-**197**)

1182 is a palindrome in two bases:

It’s 606 in BASE 14 because 6(14²) + 6(1) = 6(197) = 1182,

and 2J2 in BASE 20 (J is 19 base 10) because 2(20²) + 19(20) + 2(1) = 1182