Factors

1233 and Level 1

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Perhaps this puzzle is as difficult as a level 1 puzzle can be, but it is still not all that difficult. Nevertheless, if you can solve it, give yourself a big pat on the back.

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

Here are a few facts about the number 1233:

  • 1233 is a composite number.
  • Prime factorization: 1233 = 3 × 3 × 137, which can be written 1233 = 3² × 137
  • 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 1233 has exactly 6 factors.
  • Factors of 1233: 1, 3, 9, 137, 411, 1233
  • Factor pairs: 1233 = 1 × 1233, 3 × 411, or 9 × 137
  • Taking the factor pair with the largest square number factor, we get √1233 = (√9)(√137) = 3√137 ≈ 35.1141

Look at the numbers in this very cool but square fact about 1233:
12² + 33² = 1233

1233 is the hypotenuse of a Pythagorean triple:
792-945-1233 calculated from 2(33)(12), 33² – 12², 33² + 12²
It is also 9 times (88-105-137)

1232 Factor Cake

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1232 is divisible by 2 because it’s even.
It’s divisible by 4 because 32 is divisible by 4
It’s divisible by 8 because 232 is divisible by 8.
It also happens to be divisible by 16 and by 7.
It’s divisible by 11 because 1 – 2 + 3 – 2 = 0

1232 makes a delicious-looking factor cake:

From the factor cake, we see that 2 · 2 · 2 · 2 · 7 · 11 = 1232.

Here’s more about the number 1232:

  • 1232 is a composite number.
  • Prime factorization: 1232 = 2 × 2 × 2 × 2 × 7 × 11, which can be written 1232 = 2⁴ × 7 × 11
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 1232 has exactly 20 factors.
  • Factors of 1232: 1, 2, 4, 7, 8, 11, 14, 16, 22, 28, 44, 56, 77, 88, 112, 154, 176, 308, 616, 1232
  • Factor pairs: 1232 = 1 × 1232, 2 × 616, 4 × 308, 7 × 176, 8 × 154, 11 × 112, 14 × 88, 16 × 77, 22 × 56, or 28 × 44
  • Taking the factor pair with the largest square number factor, we get √1232 = (√16)(√77) = 4√77 ≈ 35.09986

The odd prime factors of 1232 are 7 and 11.
OEIS.org informs us that (7 × 8 × 9 × 10 × 11) / (7 + 8 + 9 + 10 + 11) = 1232

1231 Mystery Level Puzzle

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For almost all the sets of clues in this puzzle, there is more than one permissible common factor. That makes the puzzle a little tricky, but with care, you can still solve it using logic and your knowledge of the basic 10×10 multiplication table. Good luck!

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

Now I’ll tell you a little bit about the number 1231:

  • 1231 is a prime number.
  • Prime factorization: 1231 is prime.
  • The exponent of prime number 1231 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1231 has exactly 2 factors.
  • Factors of 1231: 1, 1231
  • Factor pairs: 1231 = 1 × 1231
  • 1231 has no square factors that allow its square root to be simplified. √1231 ≈ 35.08561

How do we know that 1231 is a prime number? If 1231 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1231 ≈ 35.1. Since 1231 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1231 is a prime number.

1231 is a palindrome in a couple of bases:
It’s A1A in BASE 11 because 10(11²) + 1(11) + 10(1) = 1231, and
it’s 1B1 in BASE 30 because 1(30²) + 11(30) + 1(1) = 1231

1230 Mystery

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Is this mystery level puzzle easy or difficult? The only way to know for sure is to start filling in the factors. Don’t guess and check. Use logic to find its unique solution!

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

Here is some information about the number 1230:

1230 ends with a zero so it is divisible by 2 and 5.
It is a number formed by three consecutive numbers and a zero so it is divisible by 3.

  • 1230 is a composite number.
  • Prime factorization: 1230 = 2 × 3 × 5 × 41
  • 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 1230 has exactly 16 factors.
  • Factors of 1230: 1, 2, 3, 5, 6, 10, 15, 30, 41, 82, 123, 205, 246, 410, 615, 1230
  • Factor pairs: 1230 = 1 × 1230, 2 × 615, 3 × 410, 5 × 246, 6 × 205, 10 × 123, 15 × 82, or 30 × 41
  • 1230 has no square factors that allow its square root to be simplified. √1230 ≈ 35.07136

1230 is the hypotenuse of four Pythagorean triples:
270-1200-1230 which is 30 times (9-40-41)
504-1122-1230 which is 6 times (84-187-205)
738-984-1230 which is (3-4-5) times 246
798-936-1230 which is 6 times (133-156-205)

 

1229 and Level 6

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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.

 

1228 and Level 5

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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 = 2² × 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

 

1227 and Level 4

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I’m confident you know a common factor of 42 and 60 for which ALL the factors involved are numbers from 1 to 10. That’s all you need to know to start this puzzle. Go ahead, give it a try!

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

Here is some information about the number 1227:

  • 1227 is a composite number.
  • Prime factorization: 1227 = 3 × 409
  • 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 1227 has exactly 4 factors.
  • Factors of 1227: 1, 3, 409, 1227
  • Factor pairs: 1227 = 1 × 1227 or 3 × 409
  • 1227 has no square factors that allow its square root to be simplified. √1227 ≈ 35.02856

1227 is the hypotenuse of a Pythagorean triple:
360-1173-1227 which is 3 times (120-391-409)

 

1226 Happy Birthday to My Sister, Sue

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I don’t make puzzles bigger than 12 × 12 very often, but I decided to make this one, a 17 × 17 Mystery Level for my sister’s birthday. I know she can solve smaller ones without any problems, so I wanted to give her a challenge. Happy birthday, Sue. I hope you have a great day and enjoy solving this one.

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

Note that with a bigger table there are several more possible common factors:

Is 4, 8, or 16 the common factor needed for 64 and 32 or for 16 and 48?
Is 7 or 14 the common factor needed for 14 and 70?
Is 6, 10, or 15 the common factor needed for 60 and 90?

As always there is only one solution. The table below will help anyone not familiar with some of the lesser known multiplication facts needed to solve the puzzle.

Now I’ll share some information about the number 1226:

  • 1226 is a composite number.
  • Prime factorization: 1226 = 2 × 613
  • 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 1226 has exactly 4 factors.
  • Factors of 1226: 1, 2, 613, 1226
  • Factor pairs: 1226 = 1 × 1226 or 2 × 613
  • 1226 has no square factors that allow its square root to be simplified. √1226 ≈ 35.01428

35² + 1² = 1226

1226 is the hypotenuse of a Pythagorean triple:
70-1224-1226 calculated from 2(35)(1), 35² – 1², 35² + 1²

How I Knew Immediately that a Factor Pair of 1224 is . . .

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Check Out This Pattern!

12 = 3 × 4 and 24 is one less than 25. Those two facts helped me to know right away that 35² = 1225 and 34 × 36 = 1224. Study the patterns in the chart below and you will likely be able to remember all of the multiplication facts listed in it!

a² – b² = (a – b)(a + b)
You may remember how to factor that from algebra class. Here when b = 1, it has a practical application that can allow you to amaze your friends and family with your mental calculating abilities!

I’ve only typed a small part of that infinite pattern chart. For example, if you know that 19 × 20 = 380, then you can also know that 195² = 38025 and 194 × 196 = 38024.

Also because of that chart, I know that 3.5² = 12.25 and 3.4 × 3.6 = 12.24
(Also (3½)² = 12¼, but 2½  × 4½ = 11¼ because 3-1 = 2, 3+1 = 4, 12-1 = 11
thus 2.5 × 4.5 = 11.25 and 2½  × 4½ = 11¼)

You could also let b = 2 so b² = 4. Then 25 – 4 = 21, and you could know facts like
33 × 37 = 1221 or 193 ×  197 = 38021.

I hope you have a wonderful time being a calculating genius!

A Factor Tree for 1224:

When a number has many factors, I often will make a forest of factor trees for that number, but today I just want us to enjoy this one tree for 34 × 36 = 1224.

Factors of 1224:

Now I’ll share some other facts about the number 1224:

  • 1224 is a composite number.
  • Prime factorization: 1224 = 2 × 2 × 2 × 3 × 3 × 17, which can be written 1224 = 2³ × 3² × 17
  • The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (3 + 1)(2 + 1)(1 + 1) = 4 × 3 × 2 = 24. Therefore 1224 has exactly 24 factors.
  • Factors of 1224: 1, 2, 3, 4, 6, 8, 9, 12, 17, 18, 24, 34, 36, 51, 68, 72, 102, 136, 153, 204, 306, 408, 612, 1224
  • Factor pairs: 1224 = 1 × 1224, 2 × 612, 3 × 408, 4 × 306, 6 × 204, 8 × 153, 9 × 136, 12 × 102, 17 × 72, 18 × 68, 24 × 51 or 34 × 36
  • Taking the factor pair with the largest square number factor, we get √1224 = (√36)(√34) = 6√34 ≈ 34.98571

Sum-Difference Puzzle:

1224 has twelve factor pairs. One of the factor pairs adds up to 145, and a different one subtracts to 145. If you can identify those factor pairs, then you can solve this puzzle!

If finding a sum and a difference equalling 3-digit 145 is too challenging, the chart below will be helpful.

More about the Number 1224:

1224 is also the sum of two squares:
30² + 18² = 1224

1224 is the hypotenuse of a Pythagorean triple:
576-1080-1224 which is (8-15-17) times 72
That triple can also be calculated from 30² – 18², 2(30)(18), 30² + 18²

293 + 307 + 311 + 313 = 1224 making 1224 the sum of four consecutive prime numbers.

1223 and Level 3

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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.