factor pairs

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

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)

 

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.

1222 and Level 2

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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

1221 and Level 1

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This puzzle is like a multiplication table with its factors in a different order. Can you figure out where the factors from 1 to 10 go in both the first column and the top row? Afterward, can you correctly fill in every cell of this mixed-up multiplication table?

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

Let me share some facts about the number 1221:

  • 1221 is a composite number.
  • Prime factorization: 1221 = 3 × 11 × 37
  • 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 1221 has exactly 8 factors.
  • Factors of 1221: 1, 3, 11, 33, 37, 111, 407, 1221
  • Factor pairs: 1221 = 1 × 1221, 3 × 407, 11 × 111, or 33 × 37
  • 1221 has no square factors that allow its square root to be simplified. √1221 ≈ 34.94281

1 × 11 × 111 = 1221

1221 is the sum of five consecutive prime numbers:
233 + 239 + 241 + 251 + 257 = 1221

1221 is the hypotenuse of a Pythagorean triple:
396-1155-1221 which is 33 times (12-35-37)

Not only is 1221 a palindrome in base 10 but look at it in these other bases:
It’s 14341 in BASE 5,
5353 in BASE 6,
272 in BASE 23, and
it’s XX in BASE 36 because 33(36) + 33(1) = 33(37) = 1221

1220 Challenge Puzzle

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The last challenge puzzle was particularly difficult. This one won’t be nearly as bad. Try it and see if you can figure it out!

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

Here are a few facts about the number 1220:

  • 1220 is a composite number.
  • Prime factorization: 1220 = 2 × 2 × 5 × 61, which can be written 1220 = 2² × 5 × 61
  • 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 1220 has exactly 12 factors.
  • Factors of 1220: 1, 2, 4, 5, 10, 20, 61, 122, 244, 305, 610, 1220
  • Factor pairs: 1220 = 1 × 1220, 2 × 610, 4 × 305, 5 × 244, 10 × 122, or 20 × 61
  • Taking the factor pair with the largest square number factor, we get √1220 = (√4)(√305) = 2√305 ≈ 34.9285

1220 is the sum of consecutive prime numbers: 607 + 613 = 1220

1220 is the sum of two squares two different ways:
32² + 14² = 1220
34² + 8² = 1220

1220 is the hypotenuse of four Pythagorean triples:
220-1200-1220 which is 20 times (11-60-61)
544-1092-1220 calculated from 2(34)( 8), 34² – 8², 34² + 8²
and is also 4 times (136-273-305)
828-896-1220 calculated from 32² – 14², 2(32)(14), 32² + 14²
and is also 4 times (207-224-305)
732-976-1220 which is (3-4-5) times 244

1219 is a Centered Triangular Number

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If you look at an ordered list of centered triangular numbers, 1219 will be the 29th number on the list.

Study this graphic to see why:

Here’s more about the number 1219:

  • 1219 is a composite number.
  • Prime factorization: 1219 = 23 × 53
  • 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 1219 has exactly 4 factors.
  • Factors of 1219: 1, 23, 53, 1219
  • Factor pairs: 1219 = 1 × 1219 or 23 × 53
  • 1219 has no square factors that allow its square root to be simplified. √1219 ≈ 34.91418

1219 is the sum of the thirteen prime numbers from 67 to 127.

1219 is also the hypotenuse of a Pythagorean triple:
644-1035-1219 which is 23 times (28-45-53)