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.

 

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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

1225 is a Triangular Number, a Perfect Square, and . . .

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1 and 36 are both triangular numbers and square numbers. The next number that can make the same claim is 1225. Why is this so?

1225 is a triangular number that just happens to be the sum of two consecutive triangular numbers. Two consecutive triangular numbers can always be made into a perfect square. Here’s a gif that tries to illustrate these facts about 1225:

1225 Perfect Square and Triangular Number

GIFs like this at MakeaGif

1225 is the 49th triangular number because 49(50)/2 =1225

It is the 35th square number because 35² = 1225

That it is both a triangular number AND a square number is pretty remarkable. But guess what? 1225 is ALSO a hexagonal number.

1225 is the 25th hexagonal number because (25)(2·25 – 1)= 1225
In fact, 1225 is the smallest number greater than 1 that is a triangle, a square, AND a hexagon! (Yeah, 1 is also all three and a whole more, but does 1 dot REALLY look like a triangle, a square, a hexagon and everything else all at the same time?)
All of these facts are great reasons to get very excited about the number 1225. Here are some more facts about this number:
  • 1225 is a composite number.
  • Prime factorization: 1225 = 5 × 5 × 7 × 7, which can be written 1225 = 5²× 7²
  • The exponents in the prime factorization are 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1) = 3 × 3 = 9. Therefore 1225 has exactly 9 factors.
  • Factors of 1225: 1, 5, 7, 25, 35, 49, 175, 245, 1225
  • Factor pairs: 1225 = 1 × 1225, 5 × 245, 7 × 175, 25 × 49, or 35 × 35
  • 1225 is a perfect square. √1225 = 35
Not only is 1225 a triangular number that is the sum of two other triangular numbers, but
1225 is also a square that is the sum of two other squares!
28² + 21² = 35² = 1225

1225 is the hypotenuse of two Pythagorean triples:
735-980-1225 which is (3-4-5) times 245
343-1176-1225 calculated from 28² – 21², 2(28)(21), 28² + 21²,
but it is also (7-24-25) times 49

1225 looks like a square in some other bases:
It’s 441 in BASE 17,
169 in BASE 32,
144 in BASE 33,
121 in BASE 34, and
100 in BASE 35

Stetson.edu reports that 1225 is the smallest number that can be written as the sum of four cubes three different ways. It looks like these are the three ways:
10³ + 6 ³ + 2³ + 1³ = 1225
9³ + 6³ + 6³ + 4³ = 1225
8³ + 7³ + 7³ + 3³ = 1225

1225 is, indeed, the smallest number that has all the special properties listed above!

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

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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!

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

When a number has so 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.

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.

 

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