factor pairs

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)

 

1198 Challenge Puzzle

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You can solve this Find the Factors 1 – 10 puzzle if you use logic. Guessing and checking will likely only frustrate you. Go ahead and give logic a try!

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Now I’ll share some facts about the number 1198:

  • 1198 is a composite number.
  • Prime factorization: 1198 = 2 × 599
  • 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 1198 has exactly 4 factors.
  • Factors of 1198: 1, 2, 599, 1198
  • Factor pairs: 1198 = 1 × 1198 or 2 × 599
  • 1198 has no square factors that allow its square root to be simplified. √1198 ≈ 34.61214

1198 is also palindrome 262 in BASE 23

1197 Mystery Level

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The first few moves needed to solve this puzzle might not be too hard, but soon enough it might get a bit tougher. Nevertheless, its one solution can be found using logic and an ordinary 12×12 multiplication table.

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Here are facts about the number 1197:

  • 1197 is a composite number.
  • Prime factorization: 1197 = 3 × 3 × 7 × 19, which can be written 1197 = 3² × 7 × 19
  • 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 1197 has exactly 12 factors.
  • Factors of 1197: 1, 3, 7, 9, 19, 21, 57, 63, 133, 171, 399, 1197
  • Factor pairs: 1197 = 1 × 1197, 3 × 399, 7 × 171, 9 × 133, 19 × 63, or 21 × 57
  • Taking the factor pair with the largest square number factor, we get √1197 = (√9)(√133) = 3√133 ≈ 34.59769

1197 is the sum of these eleven consecutive prime numbers:
83 + 89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 1197

1197 looks interesting to me when it is written in some other bases:
It’s 3330 in BASE 7 because 3(7³ + 7² + 7¹) = 3(399) = 1197,
and it’s 2255 in BASE 8.
It’s 999 in BASE 11, because 9(11² + 11 + 1) = 9(133) = 1197,
and it’s 1K1 in BASE 26 (K is 20 base 10)

 

1196 and Level 6

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In this puzzle, the permissible common factors of 48 and 72 are 6, 8, and 12. For clues 8 and 16, you can choose from common factors 2, 4, or 8. Which choices will make the puzzle work? I’m not telling, but I promise that the entire puzzle can be solved using logic and a basic knowledge of a 12×12 multiplication table. There is only one solution.

Print the puzzles or type the solution in this excel file: 12 factors 1187-1198

Here are some facts about the number 1196:

  • 1196 is a composite number.
  • Prime factorization: 1196 = 2 × 2 × 13 × 23, which can be written 1196 = 2² × 13 × 23
  • 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 1196 has exactly 12 factors.
  • Factors of 1196: 1, 2, 4, 13, 23, 26, 46, 52, 92, 299, 598, 1196
  • Factor pairs: 1196 = 1 × 1196, 2 × 598, 4 × 299, 13 × 92, 23 × 52, or 26 × 46
  • Taking the factor pair with the largest square number factor, we get √1196 = (√4)(√299) = 2√299 ≈ 34.58323

1196 is the hypotenuse of a Pythagorean triple:
460-1104-1196 which is (5-12-13) times 92

1196 is a palindrome in three different bases:
It’s 14241 in BASE 5,
838 in BASE 12, and
616 in BASE 14