Prime factorization

1209 Mystery Level

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How easy or difficult is this mystery level puzzle? That’s part of the mystery! Once you solve it, you will know, and you don’t have to tell let anybody else know.

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

  • 1209 is a composite number.
  • Prime factorization: 1209 = 3 × 13 × 31
  • 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 1209 has exactly 8 factors.
  • Factors of 1209: 1, 3, 13, 31, 39, 93, 403, 1209
  • Factor pairs: 1209 = 1 × 1209, 3 × 403, 13 × 93, or 31 × 39
  • 1209 has no square factors that allow its square root to be simplified. √1209 ≈ 34.7707

Did you notice the pattern in the factors?  3×13×31 = 1209
39 and 93 are two of its factors, as well!

1209 is also the hypotenuse of a Pythagorean triple:
465-1116-1209 which is (5-12-13) times 93

1208 Mystery Level

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The factors in the multiplication table puzzle below are not in the usual order. Can you figure out where each factor from 1 to 10 belongs in both the first column and the top row?

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

Here are a few facts about the number 1208:

  • 1208 is a composite number.
  • Prime factorization: 1208 = 2 × 2 × 2 × 151, which can be written 1208 = 2³ × 151
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 1208 has exactly 8 factors.
  • Factors of 1208: 1, 2, 4, 8, 151, 302, 604, 1208
  • Factor pairs: 1208 = 1 × 1208, 2 × 604, 4 × 302, or 8 × 151
  • Taking the factor pair with the largest square number factor, we get √1208 = (√4)(√302) = 2√302 ≈ 34.75629

1208 is also the sum of consecutive prime numbers:
601 + 607 = 1208

1207 The Risks of Tearing Down Walls

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When we moved into our house twenty-five years ago, the professional movers somehow managed to get our bedroom dresser around our living room wall and upstairs where we wanted it. Nevertheless, after watching their tricky maneuvering, I knew that dresser wasn’t ever leaving upstairs so long as that wall remained.

Over the last sixteen years, my husband has longed for a new bedroom set. Every time he’s brought it up, I’ve pointed to that living room wall. Sometimes walls keep us from doing what we want to do. Taking down a wall can be risky, and it can be quite messy. In our case, when the wall came down, we also got a hole in our ceiling from one joist to another. Our air conditioning intake vent was in that wall so it had to be moved and holes in the floor needed to be repaired. On both ends of the wall, we had light switches that required moving. This seemingly simple wall removal required us to hire an electrician, a HVAC expert, and a drywall expert. Here’s how it looks today right after it was removed:

Not taking down walls also has risks. We decided that the benefits and risks of taking down the wall outweighed those of keeping it.

Many people have put up an anti-math wall in their lives.

Is such a wall keeping you from doing something you really want to do? Is it keeping you from getting a college degree or pursuing an occupation that would bring you fulfillment? Tearing down that wall may require you to hire some experts to help you patch up the holes you have in your skills. However, whatever you have to do, it is worth it if it helps you fulfill your dreams.

Sometimes if you look into what you thought was a boring topic, you might find something really interesting about it.

For example, 1207 might seem like a boring number, but I bet I can find at least one interesting fact about it:

  • 1207 is a composite number.
  • Prime factorization: 1207 = 17 × 71
  • 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 1207 has exactly 4 factors.
  • Factors of 1207: 1, 17, 71, 1207
  • Factor pairs: 1207 = 1 × 1207 or 17 × 71
  • 1207 has no square factors that allow its square root to be simplified. √1207 ≈ 34.74191

Did you notice that 17 × 71 = 1207?  Likewise 71 × 17 = 1207
Its prime factors are looking in the mirror at each other!

1207 is the sum of three consecutive prime numbers:
397 + 401 + 409 = 1207

1207 can also be written as the difference of two squares two different ways:

44² – 27² = 1207
604² – 603² = 1207

And guess what? I haven’t written everything that could be written about this number. You can actually learn more about it if you choose to break down the anti-math wall to find out more!

 

1206 and Level 6

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If you carefully study all the clues in this Level 6 puzzle and use logic, you should be able to solve the puzzle. Stick with it and you’ll succeed!

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

Here are some facts about the number 1206:

  • 1206 is a composite number.
  • Prime factorization: 1206 = 2 × 3 × 3 × 67, which can be written 1206 = 2 × 3² × 67
  • The exponents in the prime factorization are 1, 2, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1) = 2 × 3 × 2 = 12. Therefore 1206 has exactly 12 factors.
  • Factors of 1206: 1, 2, 3, 6, 9, 18, 67, 134, 201, 402, 603, 1206
  • Factor pairs: 1206 = 1 × 1206, 2 × 603, 3 × 402, 6 × 201, 9 × 134, or 18 × 67,
  • Taking the factor pair with the largest square number factor, we get √1206 = (√9)(√134) = 3√134 ≈ 34.72751

Notice that 6·201 = 1206. Not very many numbers can equal themselves by using their own digits in a different way with +, -, ×, ÷, and/or parenthesis. That fact makes 1206 only the seventeenth Friedman Number.

1206 is also the sum of the twenty prime numbers from 19 to 103.

1205 and Level 5

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These Level 5 puzzles always have at least one set of clues whose common factor can only be one number. Find it, and you’ll be able to proceed using just logic and basic multiplication facts. Have fun with it!

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

Here are some facts about the number 1205:

  • 1205 is a composite number.
  • Prime factorization: 1205 = 5 × 241
  • 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 1205 has exactly 4 factors.
  • Factors of 1205: 1, 5, 241, 1205
  • Factor pairs: 1205 = 1 × 1205 or 5 × 241
  • 1205 has no square factors that allow its square root to be simplified. √1205 ≈ 34.7131

26² + 23² = 1205
34² +  7² = 1205

1205 is the hypotenuse of FOUR Pythagorean triples:
147-1196-1205 calculated from 26² – 23², 2(26)(23), 26² + 23²
476-1107-1205 calculated from 2(34)(7), 34² –  7², 34² +  7²
600-1045-1205 which is 5 times (120-209-241)
723-964-1205 which is (3-4-5) times 241

1204 and Level 4

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Today’s puzzle looks like a giant times table with a big X in the middle. The factors for this times table are not in the usual places. Can you figure out where they all go?

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

Here are a few facts about the number 1204:

  • 1204 is a composite number.
  • Prime factorization: 1204 = 2 × 2 × 7 × 43, which can be written 1204 = 2² × 7 × 43
  • 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 1204 has exactly 12 factors.
  • Factors of 1204: 1, 2, 4, 7, 14, 28, 43, 86, 172, 301, 602, 1204
  • Factor pairs: 1204 = 1 × 1204, 2 × 602, 4 × 301, 7 × 172, 14 × 86, or 28 × 43
  • Taking the factor pair with the largest square number factor, we get √1204 = (√4)(√301) = 2√301 ≈ 34.6987

1204 is the difference of two squares two different ways:
302² – 300² = 1204
50² – 36² = 1204

1203 and Level 3

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At the top of this level 3 puzzle are two clues that will tell you where to put three of the factors needed to solve the puzzle. After you find those three clues work down looking at the clues cell by cell until you have the entire puzzle solved.

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

Here are a few facts about the number 1203:

  • 1203 is a composite number.
  • Prime factorization: 1203 = 3 × 401
  • 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 1203 has exactly 4 factors.
  • Factors of 1203: 1, 3, 401, 1203
  • Factor pairs: 1203 = 1 × 1203 or 3 × 401
  • 1203 has no square factors that allow its square root to be simplified. √1203 ≈ 34.68429

Since 1203 is only made from three consecutive numbers (1, 2, 3) and zeros, it has to be divisible by 3.

1203 is the hypotenuse of a Pythagorean triple:
120-1197-1203 which is 3(40-399-401)

1202 and Level 2

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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!

 

The factors of the hundred numbers just before 1201

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I’ve made a simple chart of the numbers from 1101 to 1200, but it’s packed with great information. It gives the prime factorization of each of those numbers and how many factors each of those numbers have. The numbers written with a pinkish hue are the ones whose square roots can be simplified. Notice that each of those numbers has an exponent in its prime factorization.

I didn’t make a horserace from the amounts of factors this time because it isn’t a very close race. Nevertheless, you can guess which number appears most often in the “Amount of Factors columns” and see if your number would have won the race.

Now I’ll share some information about the next number, 1201. Notice the last entry in the chart above. It had so many factors that there weren’t very many left for 1201 to have. . .

  • 1201 is a prime number.
  • Prime factorization: 1201 is prime.
  • The exponent of prime number 1201 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1201 has exactly 2 factors.
  • Factors of 1201: 1, 1201
  • Factor pairs: 1201 = 1 × 1201
  • 1201 has no square factors that allow its square root to be simplified. √1201 ≈ 34.65545

How do we know that 1201 is a prime number? If 1201 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1201 ≈ 34.7. Since 1201 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1201 is a prime number.

Even though it doesn’t have many factors, 1201 is still a fabulous number:

25² + 24² = 1201

1201 is the 25th Centered Square Number because 25² + 24² = 1201, and 24 and 25 are consecutive numbers:

1201 is the hypotenuse of a primitive Pythagorean triple:
49-1200-1201 calculated from 25² – 24², 2(25)(24), 25² + 24²

Here’s another way we know that 1201 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 25² + 24² = 1201 with 25 and 24 having no common prime factors, 1201 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1201 ≈ 34.7. Since 1201 is not divisible by 5, 13, 17, or 29, we know that 1201 is a prime number.

1199 and Level 1

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Here’s a puzzle that even someone just learning to multiply and divide can solve. That means you can solve it, too!

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

Here are some facts about the number 1199:

  • 1199 is a composite number.
  • Prime factorization: 1199 = 11 × 109
  • 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 1199 has exactly 4 factors.
  • Factors of 1199: 1, 11, 109, 1199
  • Factor pairs: 1199 = 1 × 1199 or 11 × 109
  • 1199 has no square factors that allow its square root to be simplified. √1199 ≈ 34.62658

1199 is the sum of the fifteen prime numbers from 47 to 109. That last one just happens to be one of its prime factors, too!

1199 is the hypotenuse of a Pythagorean triple:
660-1001-1199 which is 11 times (60-91-109)

1199 looks cool in base 10, and it’s palindrome
2F2 in BASE 21 (F is 15 base 10)