1213 and Level 3

This puzzle would be a lot tougher to solve if I didn’t put the clues in the order that I did. Just start at the top of the puzzle and work your way down the puzzle clue by clue until you get to the bottom of the puzzle and have the entire thing solved.

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

Now I’ll share a few facts about the number 1213:

  • 1213 is a prime number.
  • Prime factorization: 1213 is prime.
  • The exponent of prime number 1213 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1213 has exactly 2 factors.
  • Factors of 1213: 1, 1213
  • Factor pairs: 1213 = 1 × 1213
  • 1213 has no square factors that allow its square root to be simplified. √1213 ≈ 34.82815

How do we know that 1213 is a prime number? If 1213 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1213 ≈ 34.8. Since 1213 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1213 is a prime number.

1213 is the sum of the nine consecutive prime numbers:
109 + 113 + 127 + 131 + 137 + 139 + 149 + 151 + 157 = 1213

27² + 22² = 1213
1213 is the hypotenuse of a Pythagorean triple:
245-1188-1213 calculated from 27² – 22², 2(27)(22), 27² + 22²

Here’s another way we know that 1213 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 27² + 22² = 1213 with 27 and 22 having no common prime factors, 1213 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1213 ≈ 34.8. Since 1213 is not divisible by 5, 13, 17, or 29, we know that 1213 is a prime number.

 

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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)

1211 and Level 1

Today we are reminded that the world can be a very complicated place. Today’s puzzle isn’t the least bit complicated. Just write the numbers from 1 to 12 in the first column and the top row so that the puzzle looks like a multiplication table (but with the factors not in their usual places.) Afterward, you can fill in the rest of the table. You can do this!

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

Now I’d like to share some information about the number 1211:

  • 1211 is a composite number.
  • Prime factorization: 1211 = 7 × 173
  • 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 1211 has exactly 4 factors.
  • Factors of 1211: 1, 7, 173, 1211
  • Factor pairs: 1211 = 1 × 1211 or 7 × 173
  • 1211 has no square factors that allow its square root to be simplified. √1211 ≈ 34.79943

1211 is the sum of the seventeen prime numbers from 37 to 107.
It is also the sum of seven consecutive primes:
157 + 163 + 167 + 173 + 179 + 181 + 191 = 1211

1211 is the hypotenuse of a Pythagorean triple:
364-1155-1211 which is 7 times (52-165-173)

1210 Challenge Puzzle

Some of these Find the Factors Challenge puzzles are easier than others. This one is probably not one of the easier ones. Have fun with it anyway!

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

Now I’ll share a few facts about the number 1210:

  • 1210 is a composite number.
  • Prime factorization: 1210 = 2 × 5 × 11 × 11, which can be written 1210 = 2 × 5 × 11²
  • 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 1210 has exactly 12 factors.
  • Factors of 1210: 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605, 1210
  • Factor pairs: 1210 = 1 × 1210, 2 × 605, 5 × 242, 10 × 121, 11 × 110, or 22 × 55,
  • Taking the factor pair with the largest square number factor, we get √1210 = (√121)(√10) = 11√10 ≈ 34.78505

1210 is the hypotenuse of a Pythagorean triple:
726-968-1210 which is (3-4-5) times 242

1210 is
1122211 in BASE 3
626 in BASE 14
181 in BASE 31

1209 Mystery Level

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

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

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

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

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

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