# 1312 Fill This Boot with Candy

On the 5th of December, many children in the world prepare for a visit from Saint Nickolas by polishing their boots. Hopefully, they have been good boys or girls all year and will find those boots filled the next morning with their favorite candies. Here’s a boot-shaped puzzle for you to solve.

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

Now I’ll share some information about the number 1312:

• 1312 is a composite number.
• Primefactorization: 1312 = 2 × 2 × 2 × 2 × 2 × 41, which can be written 1312 = 2⁵ × 41
• The exponents inthe prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 1312 has exactly 12 factors.
• Factors of 1312: 1, 2, 4, 8, 16, 32, 41, 82, 164, 328, 656, 1312
• Factor pairs: 1312 = 1 × 1312, 2 × 656, 4 × 328, 8 × 164, 16 × 82, or 32 × 41
• Taking the factor pair with the largest square number factor, we get √1312 = (√16)(√82) = 4√82 ≈ 36.22154

1312 is the sum of consecutive prime numbers in two different ways:
It is the sum of the sixteen prime numbers from 47 to 113. Also,
prime numbers 653 + 659 = 1312

1312 is the sum of two squares:
36² + 4² = 1312

1312 is also the hypotenuse of a Pythagorean triple:
288-1280-1312 which is 32 times (9-40-41)

# 1303 and Level 2

Multiplication tables usually have facts up to 10 × 10 = 100 or possibly 12 × 12 = 144. Numbers like 64 and 25 appear only once in those multiplication tables. Those two clues can help you get a good start solving this level 2 puzzle. Print the puzzles or type the solution in this excel file: 10-factors-1302-1310

Now I’ll share a few facts about the number 1303:

• 1303 is a prime number.
• Prime factorization: 1303 is prime.
• The exponent of prime number 1303 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1303 has exactly 2 factors.
• Factors of 1303: 1, 1303
• Factor pairs: 1303 = 1 × 1303
• 1303 has no square factors that allow its square root to be simplified. √1303 ≈ 36.09709

How do we know that 1303 is a prime number? If 1303 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1303 ≈ 36.1. Since 1303 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1303 is a prime number. 1303 is the sum of three consecutive primes:
431 + 433 + 439 = 1303

# 1290 Multiplication Boomerang

Do multiplication and division facts seem like something you threw out long ago but still come back to hit you? Perhaps this puzzle can help you get more familiar with those facts so they won’t hurt you so much anymore. Print the puzzles or type the solution in this excel file: 12 factors 1289-1299

That was puzzle number 1290. Here are some facts about that number:

• 1290 is a composite number.
• Prime factorization: 1290 = 2 × 3 × 5 × 43
• 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 1290 has exactly 16 factors.
• Factors of 1290: 1, 2, 3, 5, 6, 10, 15, 30, 43, 86, 129, 215, 258, 430, 645, 1290
• Factor pairs: 1290 = 1 × 1290, 2 × 645, 3 × 430, 5 × 258, 6 × 215, 10 × 129, 15 × 86, or 30 × 43
• 1290 has no square factors that allow its square root to be simplified. √1290 ≈ 35.91657 1290 is the sum of two consecutive prime numbers:
643 + 647 = 1290

1290 is the hypotenuse of a Pythagorean triple:
774-1032-1290 which is (3-4-5) times 258

# 1282 and Level 2

Can you find the factors from 1 to 10 that make the twelve clues in the puzzle the correct products for this scrambled multiplication table? Print the puzzles or type the solution in this excel file: 10-factors-1281-1288

Here is some information about the number 1282:

• 1282 is a composite number.
• Prime factorization: 1282 = 2 × 641
• 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 1282 has exactly 4 factors.
• Factors of 1282: 1, 2, 641, 1282
• Factor pairs: 1282 = 1 × 1282 or 2 × 641
• 1282 has no square factors that allow its square root to be simplified. √1282 ≈ 35.80503 1282 is the sum of two squares:
29² +  21² = 1282

1282 is the hypotenuse of a primitive Pythagorean triple:
400-1218-1282 calculated from 29² –  21², 2(29)( 21), 29² +  21²

The 21, 29, and 400 above are related to another Pythagorean triple:
20-21-29 because 20² = 400, 21² = 441 and 29² = 841. Thus,
400 + 441 = 841. Pretty cool!

# 1272 and Level 2

Some of the factor pairs needed to solve this puzzle may be easier for you to find than others, but I’m sure you can still find all of them. Give it a try! Print the puzzles or type the solution in this excel file: 12 factors 1271-1280

Now I’ll write a little bit about the number 1272:

• 1272 is a composite number.
• Prime factorization: 1272 = 2 × 2 × 2 × 3 × 53, which can be written 1272 = 2³ × 3 × 53
• The exponents in the prime factorization are 3, 1, and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1)(1 + 1) = 4 × 2 × 2 = 16. Therefore 1272 has exactly 16 factors.
• Factors of 1272: 1, 2, 3, 4, 6, 8, 12, 24, 53, 106, 159, 212, 318, 424, 636, 1272
• Factor pairs: 1272 = 1 × 1272, 2 × 636, 3 × 424, 4 × 318, 6 × 212, 8 × 159, 12 × 106, or 24 × 53
• Taking the factor pair with the largest square number factor, we get √1272 = (√4)(√318) = 2√318 ≈ 35.66511 1272 is the sum of four consecutive prime numbers, and it is the sum of two consecutive prime numbers:
311 + 313 + 317 + 331 = 1272
631 + 641 = 1272

1272 is the hypotenuse of a Pythagorean triple:
672-1080-1272 which is 24 times (28-45-52)

# 1253 and Level 2

In what order should the numbers from 1 to 12 be written in the first column and also in the top row so that this puzzle works like a multiplication table? Print the puzzles or type the solution in this excel file: 12 factors 1251-1258

Now I’ll tell you a little bit about the number 1253:

• 1253 is a composite number.
• Prime factorization: 1253 = 7 × 179
• 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 1253 has exactly 4 factors.
• Factors of 1253: 1, 7, 179, 1253
• Factor pairs: 1253 = 1 × 1253 or 7 × 179
• 1253 has no square factors that allow its square root to be simplified. √1253 ≈ 35.39774 1253 is also the sum of the eleven prime numbers from 89 to 139. Do you know what those prime numbers are?

# 1244 and Level 2

If you can find the common factors of the clues in each row or column of this puzzle, then you can solve this puzzle. Be sure to only write numbers from 1 to 10 as those factors, and I’m sure you can succeed. Print the puzzles or type the solution in this excel file: 10-factors-1242-1250

Here are some facts about the number 1244:

• 1244 is a composite number.
• Prime factorization: 1244 = 2 × 2 × 311, which can be written 1244 = 2² × 311
• 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 1244 has exactly 6 factors.
• Factors of 1244: 1, 2, 4, 311, 622, 1244
• Factor pairs: 1244 = 1 × 1244, 2 × 622, or 4 × 311
• Taking the factor pair with the largest square number factor, we get √1244 = (√4)(√311) = 2√311 ≈ 35.27038 1244 is the sum of the cubes of the first four triangular numbers:
1³ + 3³ + 6³ + 10³ =1244

1244 is a palindrome in a couple of different bases:
It’s 878 in BASE 12 and
282 in BASE 23

# 1234 and Level 2

This is my 1234th post, so today’s puzzle has been given that number. Whenever I see 12:34 on a clock, I always think about my husband’s Uncle Paul who really liked noticing that time because all possible clock digits are used and the digits are in order. I also like those digits because 12 = 3 × 4. Print the puzzles or type the solution in this excel file: 12 factors 1232-1241

Here are some facts about the number 1234 some of which might surprise you:

• 1234 is a composite number.
• Prime factorization: 1234 = 2 × 617
• 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 1234 has exactly 4 factors.
• Factors of 1234: 1, 2, 617, 1234
• Factor pairs: 1234 = 1 × 1234 or 2 × 617
• 1234 has no square factors that allow its square root to be simplified. √1234 ≈ 35.12834 1234 is the sum of two squares:
35² + 3² = 1234

1234 is the hypotenuse of a Pythagorean triple:
210-1216-1234 calculated from 2(35)(3), 35² – 3², 35² + 3²
It is also times (105-608-617)

# 1222 and Level 2

Can you write the numbers from 1 to 10 in both the first column and the top row of the table below so that the given clues are the multiplication products of the factors you wrote? There is only one solution, but I am sure that you can find it. Print the puzzles or type the solution in this excel file: 10-factors-1221-1231

Now I’ll write a little bit about the number 1222:

• 1222 is a composite number.
• Prime factorization: 1222 = 2 × 13 × 47
• 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 1222 has exactly 8 factors.
• Factors of 1222: 1, 2, 13, 26, 47, 94, 611, 1222
• Factor pairs: 1222 = 1 × 1222, 2 × 611, 13 × 94, or 26 × 47
• 1222 has no square factors that allow its square root to be simplified. √1222 ≈ 34.95712 1222 is the hypotenuse of a Pythagorean triple:
470-1128-1222 which is (5-12-13) times 94

# 1212 and Level 2

I know you will find at least some of today’s puzzle to be easy, and maybe even all of it. Why not give it a try? You might just have fun solving it, too! Print the puzzles or type the solution in this excel file: 12 factors 1211-1220

When we learn about the number 1212 we get to think about twelve a lot. 1212 has twelve factors and, of course, one of them is 12.

• 1212 is a composite number.
• Prime factorization: 1212 = 2 × 2 × 3 × 101, which can be written 1212 = 2² × 3 × 101
• 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 1212 has exactly 12 factors.
• Factors of 1212: 1, 2, 3, 4, 6, 12, 101, 202, 303, 404, 606, 1212
• Factor pairs: 1212 = 1 × 1212, 2 × 606, 3 × 404, 4 × 303, 6 × 202, or 12 × 101
• Taking the factor pair with the largest square number factor, we get √1212 = (√4)(√303) = 2√303 ≈ 34.81379 1212 is the sum of the fourteen prime numbers from 59 to 113.
It is also the sum of the twelve prime numbers from 73 to 131.
Do you know what all those prime numbers are?

1212 is the hypotenuse of a Pythagorean triple:
240-1188-1212 which is 12 times (20-99-101)