1190 and Level 3

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The common factors of 108 and 120 are 1, 2, 3, 4, 6, and 12, but pick the one that will only put numbers from 1 to 12 in the first column. Then work down that first column cell by cell writing in the factors of the clues as you go. Each number from 1 to 12 must go somewhere in both the first column and the top row.

Now I’ll share some information about the number 1190:

  • 1190 is a composite number.
  • Prime factorization: 1190 = 2 × 5 × 7 × 17
  • 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 = 110. Therefore 1190 has exactly 110 factors.
  • Factors of 1190: 1, 2, 5, 7, 10, 14, 17, 34, 35, 70, 85, 119, 170, 238, 595, 1190
  • Factor pairs: 1190 = 1 × 1190, 2 × 595, 5 × 238, 7 × 170, 10 × 119, 14 × 85, 17 × 70, or 34 × 35
  • 1190 has no square factors that allow its square root to be simplified. √1190 ≈ 34.49638

Because 1190 is the product of consecutive numbers 34 and 35, we know it is the sum of the first 34 EVEN numbers. Instead of writing all of those 34 numbers, we can use a some mathematical shorthand and simply write:
2 + 4 + 6 + 8 + . . . + 64 + 66 + 68 = 1190

1190 is the hypotenuse of FOUR Pythagorean triples:
182-1176-1190 which is 14 times (13-84-85)
504-1078-1190 which is 14 times (36-77-85)
560-1050-1190 which is (8-15-17) times 70
714-952-1190 which is (3-4-5) times 238

I like the way 1190 looks in some other bases:
It’s 707 in BASE 13 because 7(13²) + 7(1) = 7(169 + 1) = 7(170) = 1190,
545 in BASE 15,
2A2 in BASE 22,
1C1 in BASE 29 (C is 12 base 10),
and Y0 in BASE 35 (Y is 34 base 10) because 34(35) = 1190

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1189 and Level 2

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Some of the clues in this level 2 puzzle were also in the level 1 puzzle earlier this week. Can you remember their common factors and figure out the common factors for the other three sets of clues?

Here are some facts about the number 1189:

  • 1189 is a composite number.
  • Prime factorization: 1189 = 29 × 41
  • 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 1189 has exactly 4 factors.
  • Factors of 1189: 1, 29, 41, 1189
  • Factor pairs: 1189 = 1 × 1189 or 29 × 41
  • 1189 has no square factors that allow its square root to be simplified. √1189 ≈ 34.48188

33² + 10² = 1189
30² + 17² = 1189

1189 is the hypotenuse of FOUR Pythagorean triples:
261-1160-1189 which is 29 times (9-40-41)
611-1020-1189 calculated from 30² – 17², 2(30)(17), 30² + 17²
660-989-1189 calculated from 2(33)(10), 33² – 10², 33² + 10²
820-861-1189 which is (20-21-29) times 41

1189 is 10010100101 in BASE 2. That’s a nice pattern.
It’s also palindrome 1H1 in base 27 ( H is 17 base 10),
and palindrome 131 in BASE 33

1188 How Many Triangles? How Many Factors?

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Manually counting ALL the triangles in the graphic below could get confusing. Try counting all the triangles that point up first then all the triangles that point down.

Since the number of triangle rows is an even number, you could just use this formula:

n(n+2)(2n+1)/8 = the number of triangles

In this case, n = 16, so the number of triangles is 16×18×33/8 = 1188.

(If the number of rows is an odd number, the formula produces a number with a decimal. In that case, just ignore the decimal and everything after it to get the number of triangles.)

Here’s some more information about the number 1188:

  • 1188 is a composite number.
  • Prime factorization: 1188 = 2 × 2 × 3 × 3 × 3 × 11, which can be written 1188 = 2² × 3³ × 11
  • The exponents in the prime factorization are 2, 3 and 1. Adding one to each and multiplying we get (2 + 1)(3 + 1)(1 + 1) = 3 × 4 × 2 = 24. Therefore 1188 has exactly 24 factors.
  • Factors of 1188: 1, 2, 3, 4, 6, 9, 11, 12, 18, 22, 27, 33, 36, 44, 54, 66, 99, 108, 132, 198, 297, 396, 594, 1188
  • Factor pairs: 1188 = 1 × 1188, 2 × 594, 3 × 396, 4 × 297, 6 × 198, 9 × 132, 11 × 108, 12 × 99, 18 × 66, 22 × 54, 27 × 44 or 33 × 36
  • Taking the factor pair with the largest square number factor, we get √1188 = (√36)(√33) = 6√33 ≈ 34.46738

1188 has MANY possible factor trees. Here are four of them:

1188 is an interesting-looking base 10 number, but it also looks interesting when it is written in some other bases:
It’s 2244 in BASE 8,
990 in BASE 11 because 9(11²) + 9(11) = 9(132) = 1188,
543 in BASE 15,
4A4 in BASE 16 (A is 10 base 10),
XX in BASE 35 (X is 33 base 10) because 33(35) + 33(1) = 33(36) = 1188,
and it’s X0 in BASE 36 because 33(36) = 1188

1187 and Level 1

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What is the biggest number that can divide all the clues in today’s puzzle without leaving a remainder? If you can answer that question, then you also know the greatest common factor of all those clues. It really is that simple. You can solve this puzzle!

Now I’ll share some information about the number 1187:

  • 1187 is a prime number.
  • Prime factorization: 1187 is prime.
  • The exponent of prime number 1187 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1187 has exactly 2 factors.
  • Factors of 1187: 1, 1187
  • Factor pairs: 1187 = 1 × 1187
  • 1187 has no square factors that allow its square root to be simplified. √1187 ≈ 34.45287

How do we know that 1187 is a prime number? If 1187 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1187 ≈ 34.5. Since 1187 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1187 is a prime number.

1187 is the sum of the nineteen prime numbers from 23 to 103.
It is also the sum of three consecutive primes:
389 + 397 + 401 = 1187

1186 Challenge Puzzle

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It shouldn’t be too hard to make your first move in this puzzle. After that, I don’t make any guarantees. You just need to write each number from 1 to 10 in each of the four boldly outlined areas so that the given clues are the products of the factors you wrote. Use logic to find all the factors and have fun doing it!

Print the puzzles or type the solution in this excel file: 10-factors-1174-1186

What have I found out about the number 1186?

  • 1186 is a composite number.
  • Prime factorization: 1186 = 2 × 593
  • 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 1186 has exactly 4 factors.
  • Factors of 1186: 1, 2, 593, 1186
  • Factor pairs: 1186 = 1 × 1186 or 2 × 593
  • 1186 has no square factors that allow its square root to be simplified. √1186 ≈ 34.43835

31² + 15² = 1186

1186 is the hypotenuse of a Pythagorean triple:
736-930-1186 calculated from 31² – 15², 2(31)(15), 31² + 15²

1186 is palindrome 989 in BASE 11

1185 Mystery Level

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The difficulty level of a Mystery Level puzzle is somewhere between fairly easy to extra hard. You won’t know how easy or how difficult it is until you give it a try. Use logic, not guessing and checking, and have fun with it!

Print the puzzles or type the solution in this excel file: 10-factors-1174-1186

What can I tell you about the number 1185?

  • 1185 is a composite number.
  • Prime factorization: 1185 = 3 × 5 × 79
  • 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 1185 has exactly 8 factors.
  • Factors of 1185: 1, 3, 5, 15, 79, 237, 395, 1185
  • Factor pairs: 1185 = 1 × 1185, 3 × 395, 5 × 237, or 15 × 79
  • 1185 has no square factors that allow its square root to be simplified. √1185 ≈ 34.42383

1185 is the hypotenuse of a Pythagorean triple:
711-948-1185 which is (3-4-5) times 237

I think 1185 looks interesting when it is written in some other bases:
It’s palindrome 102201 in BASE 4,
357 in BASE 19, and
palindrome 151 in BASE 32

1184 Mystery Level

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The sets of clues in this Mystery Level puzzle have more than one possible common factor. Don’t guess which one you should use. In each case, the right factor can be selected by using logic. Will you figure out where all the factors from 1 to 10 go?

Print the puzzles or type the solution in this excel file: 10-factors-1174-1186

Here are a few facts about the number 1184:

  • 1184 is a composite number.
  • Prime factorization: 1184 = 2 × 2 × 2 × 2 × 2 × 37, which can be written 1184 = 2⁵ × 37
  • The exponents in the prime factorization are 5 and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1) = 6 × 2 = 12. Therefore 1184 has exactly 12 factors.
  • Factors of 1184: 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, 592, 1184
  • Factor pairs: 1184 = 1 × 1184, 2 × 592, 4 × 296, 8 × 148, 16 × 74, or 32 × 37
  • Taking the factor pair with the largest square number factor, we get √1184 = (√16)(√74) = 4√74 ≈ 34.4093

1184 is the hypotenuse of a Pythagorean triple:
384-1120-1184 which is 32 times (12-35-37)

1184 looks interesting to me in a few other bases:
It’s 5252 in BASE 6,
3311 in BASE 7,
987 in BASE 11,
828 in BASE 12, and
WW in BASE 36 (W is 32 base 10)
That one is because 32(36) + 32(1) = 32(37) = 1184

1183 is the 13th Pentagonal Pyramidal Number

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1183 is the 13th pentagonal pyramidal number. Here’s an attempt to illustrate that fact. (Try to think 3-dimensionally.):

Here are some more facts about the number 1183:

  • 1183 is a composite number.
  • Prime factorization: 1183 = 7 × 13 × 13, which can be written 1183 = 7 × 13²
  • The exponents in the prime factorization are 1 and 2. Adding one to each and multiplying we get (1 + 1)(2 + 1) = 2 × 3  = 6. Therefore 1183 has exactly 6 factors.
  • Factors of 1183: 1, 7, 13, 91, 169, 1183
  • Factor pairs: 1183 = 1 × 1183, 7 × 169, or 13 × 91
  • Taking the factor pair with the largest square number factor, we get √1183 = (√169)(√7) = 13√7 ≈ 34.39476

1183 is the hypotenuse of two Pythagorean triples:
833-840-1183 which is 7 times (119-120-169)
455-1092-1183 which is (5-12-13) times 91

I like 1183 in a couple of different bases:
It’s palindrome 1121211 in BASE 3 because
3⁶ + 3⁵ + 2(3⁴) + 3³ + 2(3²) + 3 + 1 = 1183,
and it’s 700 in BASE 13 because 7(13²) = 1183

 

1182 A Unique Solution

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Every Find the Factors puzzle I make has a unique solution. That fact is central to the logic needed to begin this particular puzzle. I hope it frustrates you a little, but not too much. Then when you finally solve it, it will be so much sweeter!

Print the puzzles or type the solution in this excel file: 10-factors-1174-1186

Here are some facts about the number 1182:

  • 1182 is a composite number.
  • Prime factorization: 1182 = 2 × 3 × 197
  • 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 1182 has exactly 8 factors.
  • Factors of 1182: 1, 2, 3, 6, 197, 394, 591, 1182
  • Factor pairs: 1182 = 1 × 1182, 2 × 591, 3 × 394, or 6 × 197
  • 1182 has no square factors that allow its square root to be simplified. √1182 ≈ 34.38023

1182 is the hypotenuse of a Pythagorean triple:
168-1170-1182 which is 6 times (28-195-197)

1182 is a palindrome in two bases:
It’s 606 in BASE 14 because 6(14²) + 6(1) = 6(197) = 1182,
and 2J2 in BASE 20 (J is 19 base 10) because 2(20²) + 19(20) + 2(1) = 1182

1181 and Level 5

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Some parts of this puzzle are easier than others, but it all still a lot of fun! Give it a try and enjoy yourself!

Print the puzzles or type the solution in this excel file: 10-factors-1174-1186

Now I’ll share some facts about the number 1181:

  • 1181 is a prime number.
  • Prime factorization: 1181 is prime.
  • The exponent of prime number 1181 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1181 has exactly 2 factors.
  • Factors of 1181: 1, 1181
  • Factor pairs: 1181 = 1 × 1181
  • 1181 has no square factors that allow its square root to be simplified. √1181 ≈ 34.36568

How do we know that 1181 is a prime number? If 1181 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1181 ≈ 34.4. Since 1181 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1181 is a prime number.

34² + 5² = 1181

1181 is the hypotenuse of a Pythagorean triple:
340-1131-1181 calculated from 2(34)( 5), 34² – 5², 34² + 5²

Here’s another way we know that 1181 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 34² + 5² = 1181 with 34 and 5 having no common prime factors, 1181 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1181 ≈ 34.4. Since 1181 is not divisible by 5, 13, 17, or 29, we know that 1181 is a prime number.

1181 is also palindrome 353 in BASE 19
because 3(19²) + 5(19) + 3(1) = 1181