If you start at the top of this Christmas tree puzzle and work your way down, you can find all the factors that make this puzzle work like a multiplication table.

# Level 3 Puzzle

# 1313 Virgács and St. Nickolas Day

6 December is Saint Nickolas Day. Children in Hungary and other places in Europe wake up to find candy and virgács in their boots. You can read more about this wonderful tradition in Jön a Mikulás (Santa is Coming) or Die Feier des Weihnachtsmanns (The Celebration of Santa Claus). Today’s puzzle represents the virgács given to children who have been even the least bit naughty during the current year.

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

Now I’ll write a little bit about the number 1313:

- 1313 is a composite number.
- Prime factorization: 1313 = 13 × 101
- 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 1313 has exactly 4 factors.
- Factors of 1313: 1, 13, 101, 1313
- Factor pairs: 1313 = 1 × 1313 or 13 × 101
- 1313 has no square factors that allow its square root to be simplified. √1313 ≈ 36.23534

1313 is the sum of consecutive prime numbers three different ways:

It is the sum of the twenty-one prime numbers from 19 to 107.

It is the sum of eleven consecutive primes:

97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 = 1313,

and it is the sum of seven consecutive prime numbers:

173 + 179 + 181 + 191 + 193 + 197 + 199 = 1313

1313 is the sum of two squares two different ways:

32² + 17² = 1313

28² + 23² = 1313

1313 is the hypotenuse of FOUR Pythagorean triples:

255-1288-1313 calculated from 28² – 23², 2(28)(23), 28² + 23²

260-1287-1313 which is **13** times (20-99-**101**)

505-1212-1313 which is (5-12-**13**) times **101**

735-1088-1313 calculated from 32² – 17², 2(32)(17), 32² + 17²

# 1304 and Level 3

Which factor pairs of 45 and 18 have only numbers from 1 to 10 in them? Answer that question, put the factors in the appropriate places, and then work your way down this level 3 puzzle cell by cell until you’ve solved it.

Print the puzzles or type the solution in this excel file: 10-factors-1302-1310

Here are a few facts about the number 1304:

- 1304 is a composite number.
- Prime factorization: 1304 = 2 × 2 × 2 × 163, which can be written 1304 = 2³ × 163
- 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 1304 has exactly 8 factors.
- Factors of 1304: 1, 2, 4, 8, 163, 326, 652, 1304
- Factor pairs: 1304 = 1 × 1304, 2 × 652, 4 × 326, or 8 × 163
- Taking the factor pair with the largest square number factor, we get √1304 = (√4)(√326) = 2√326 ≈ 36.11094

1304 is the difference of two squares two ways:

327² – 325² = 1304

165² – 161² = 1304

# 1291 and Level 3

If you think of the common factors of 25 and 55, then you have actually started to solve this puzzle! Start at the top of the puzzle and work your way down. Try it!

Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

Here are some facts about the number 1291

- 1291 is a prime number.
- Prime factorization: 1291 is prime.
- The exponent of prime number 1291 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1291 has exactly 2 factors.
- Factors of 1291: 1, 1291
- Factor pairs: 1291 = 1 × 1291
- 1291 has no square factors that allow its square root to be simplified. √1291 ≈ 35.93049

**How do we know that ****1291**** is a prime number?** If 1291 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1291 ≈ 35.9. Since 1291 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1291 is a prime number.

1291 is also palindrome 1D1 in BASE 30 because 30² + 13(30) + 1 = 1291

# 1283 and Level 3

What is the greatest common factor of 28 and 63? If you know, then you can probably figure out this puzzle!

Print the puzzles or type the solution in this excel file: 10-factors-1281-1288

Here are some facts about the number 1283:

- 1283 is a prime number.
- Prime factorization: 1283 is prime.
- The exponent of prime number 1283 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1283 has exactly 2 factors.
- Factors of 1283: 1, 1283
- Factor pairs: 1283 = 1 × 1283
- 1283 has no square factors that allow its square root to be simplified. √1283 ≈ 35.81899

**How do we know that ****1283**** is a prime number?** If 1283 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1283 ≈ 35.8. Since 1283 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1283 is a prime number.

1283 is the sum of the seventeen prime numbers from 41 to 109,

AND it is the sum of the thirteen primes from 71 to 131.

# 1273 and Level 3

What’s the only common factor or 12 and 11? Write factors of 12 and 11 where they belong on the puzzle below. Then starting back up at the top of the puzzle, go down the puzzle writing the appropriate factors cell by cell until you’re done. You can do this!

Print the puzzles or type the solution in this excel file: 12 factors 1271-1280

Here’s some information about the number 1273:

- 1273 is a composite number.
- Prime factorization: 1273 = 19 × 67
- 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 1273 has exactly 4 factors.
- Factors of 1273: 1, 19, 67, 1273
- Factor pairs: 1273 = 1 × 1273 or 19 × 67
- 1273 has no square factors that allow its square root to be simplified. √1273 ≈ 35.67913

1273 is also palindrome 10011111001 in BASE 2.

# 1254 and Level 3

Find the common factor of 8 and 80 so that only numbers from 1 to 12 will be put in the top row of this multiplication table puzzle. Then work down row by row writing the factors of each clue so that the numbers from 1 to 12 appear only once in both the first column and the top row. You can do this!

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

Here are a few facts about the post number, 1254:

- 1254 is a composite number.
- Prime factorization: 1254 = 2 × 3 × 11 × 19
- The exponents in the prime factorization are 1, 1, 1, and 1. Adding one to each and multiplying we get (1 + 1)(1 + 1)(1 + 1)(1 + 1) = 2 × 2 × 2 × 2 = 16. Therefore 1254 has exactly 16 factors.
- Factors of 1254: 1, 2, 3, 6, 11, 19, 22, 33, 38, 57, 66, 114, 209, 418, 627, 1254
- Factor pairs: 1254 = 1 × 1254, 2 × 627, 3 × 418, 6 × 209, 11 × 114, 19 × 66, 22 × 57, or 33 × 38
- 1254 has no square factors that allow its square root to be simplified. √1254 ≈ 35.41186

1254 is the sum of the twenty-four prime numbers from 7 to 103. Do you know what those prime numbers are?

# 1245 and Level 3

If you know the greatest common factor of 15 and 20, then you can begin to solve this puzzle. Since this is a level 3 puzzle, look at the clues starting at the top of the puzzle and work your way down, writing in the factors as you go. You can do this!

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

Now I’ll share some facts about the number 1245:

- 1245 is a composite number.
- Prime factorization: 1245 = 3 × 5 × 83
- 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 1245 has exactly 8 factors.
- Factors of 1245: 1, 3, 5, 15, 83, 249, 415, 1245
- Factor pairs: 1245 = 1 × 1245, 3 × 415, 5 × 249, or 15 × 83
- 1245 has no square factors that allow its square root to be simplified. √1245 ≈ 35.28456

1245 is also the hypotenuse of a Pythagorean triple:

747-996-1245 which is (3-4-**5**) times **249**

# 1235 and Level 3

Do you know the greatest common factor of 28 and 35? If you do, then you can solve this puzzle by writing each number from 1 to 12 in both the first column and the top row. Since this is a level 3 puzzle, you can begin with the clues at the top of the puzzle and work your way down cell by cell. Good luck!

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

Now I’ll share some facts about the number 1235:

- 1235 is a composite number.
- Prime factorization: 1235 = 5 × 13 × 19
- 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 1235 has exactly 8 factors.
- Factors of 1235: 1, 5, 13, 19, 65, 95, 247, 1235
- Factor pairs: 1235 = 1 × 1235, 5 × 247, 13 × 95, or 19 × 65
- 1235 has no square factors that allow its square root to be simplified. √1235 ≈ 35.14257

1235 is the hypotenuse of FOUR Pythagorean triples:

304-1197-1235 which is **19** times (16-63-**65**)

741-988-1235 which is (3-4-**5**) times **247**

627-1064-1235 which is **19** times (33-56-**65**)

475-1140-1235 which is (5-12-**13**) times **95**

# 1223 and Level 3

If you’ve been too anxious to try solving a level 3 puzzle in the past, you have no excuse for not trying this one. This might be the easiest level 3 puzzle I’ve ever published. Just write the factors for 40 and 48 in the proper cells, then work your way down the puzzle writing only numbers from 1 to 10 in the first column and the top row. Seriously, you can do this one!

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

Now I’ll write a little about the number 1223:

- 1223 is a prime number.
- Prime factorization: 1223 is prime.
- The exponent of prime number 1223 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1223 has exactly 2 factors.
- Factors of 1223: 1, 1223
- Factor pairs: 1223 = 1 × 1223
- 1223 has no square factors that allow its square root to be simplified. √1223 ≈ 34.97142

**How do we know that ****1223**** is a prime number?** If 1223 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1223 ≈ 34.97. Since 1223 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1223 is a prime number.

1223 is the sum of the twenty-one prime numbers from 17 to 103.