1322 Christmas Star

The first Christmas Star led the wise men to find the Baby Jesus.

This Christmas star can lead you to a better knowledge of all the facts in a basic 1 to 10 multiplication table.

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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 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)

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)

1202 and Level 2

I am certain that you can fill in the numbers 1 to 10 one time in both the top row and the first column so that this puzzle can become a multiplication table. All you have to do is give it an honest try.

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

Now I’ll write a few things about the number 1202:

  • 1202 is a composite number.
  • Prime factorization: 1202 = 2 × 601
  • 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 1202 has exactly 4 factors.
  • Factors of 1202: 1, 2, 601, 1202
  • Factor pairs: 1202 = 1 × 1202 or 2 × 601
  • 1202 has no square factors that allow its square root to be simplified. √1202 ≈ 34.66987

29² + 19² = 1202

1202 is the hypotenuse of a Pythagorean triple:
480-1102-1202 calculated from 29² – 19², 2(29)(19), 29² + 19²

2(24² + 5²) = 2(601) = 1202 so that Pythagorean triple can also be calculated from
2(2)(24)(5), 2(24² – 5²), 2(24² + 5²)

Try out both ways to get the triple!

 

1189 and Level 2

Some of the clues in this level 2 puzzle were also in the level 1 puzzle earlier this week. Can you remember their common factors and figure out the common factors for the other three sets of clues?

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

Here are some facts about the number 1189:

  • 1189 is a composite number.
  • Prime factorization: 1189 = 29 × 41
  • 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 1189 has exactly 4 factors.
  • Factors of 1189: 1, 29, 41, 1189
  • Factor pairs: 1189 = 1 × 1189 or 29 × 41
  • 1189 has no square factors that allow its square root to be simplified. √1189 ≈ 34.48188

33² + 10² = 1189
30² + 17² = 1189

1189 is the hypotenuse of FOUR Pythagorean triples:
261-1160-1189 which is 29 times (9-40-41)
611-1020-1189 calculated from 30² – 17², 2(30)(17), 30² + 17²
660-989-1189 calculated from 2(33)(10), 33² – 10², 33² + 10²
820-861-1189 which is (20-21-29) times 41

1189 is 10010100101 in BASE 2. That’s a nice pattern.
It’s also palindrome 1H1 in base 27 ( H is 17 base 10),
and palindrome 131 in BASE 33