1186 Challenge Puzzle

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

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1185 Mystery Level

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

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

 

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

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

1180 How Are They the Same? How Are They Different?

I liked the vintage buggies at Sosto Museum Village so much that I had to snap a picture and share them with you.

How are they the same? How are they different? When you answer those questions, you are doing mathematics!

You could write the answers to those questions in a Venn Diagram. Then it would be more obvious that you are thinking mathematically!

The number 1180 is like some other numbers but different than others. Here are some facts about it:

  • 1180 is a composite number.
  • Prime factorization: 1180 = 2 × 2 × 5 × 59, which can be written 1180 = 2² × 5 × 59
  • 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 1180 has exactly 12 factors.
  • Factors of 1180: 1, 2, 4, 5, 10, 20, 59, 118, 236, 295, 590, 1180
  • Factor pairs: 1180 = 1 × 1180, 2 × 590, 4 × 295, 5 × 236, 10 × 118, or 20 × 59
  • Taking the factor pair with the largest square number factor, we get √1180 = (√4)(√295) = 2√295 ≈ 34.35113

1180 is the hypotenuse of a Pythagorean triple:
708-944-1180 which is (3-4-5) times 236

1180 is the sum of two consecutive prime numbers: 587 + 593 = 1180

1180 is palindrome 1551 in BASE 9 because 1(9³) + 5(9²) + 5(9) + 1(1) = 1180

 

 

1179 and Level 4

80 and 16 have just one common factor that will put only numbers from 1 to 10 in the first column and in the top row. Put those factors where they belong and use logic to figure out where to put the rest.

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

Now I tell you what I’ve learned about the number 1179:

  • 1179 is a composite number.
  • Prime factorization: 1179 = 3 × 3 × 131, which can be written 1179 = 3² × 131
  • 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 1179 has exactly 6 factors.
  • Factors of 1179: 1, 3, 9, 131, 393, 1179
  • Factor pairs: 1179 = 1 × 1179, 3 × 393, or 9 × 131
  • Taking the factor pair with the largest square number factor, we get √1179 = (√9)(√131) = 3√131 ≈ 34.33657

I like the way 1179 looks in a couple other bases:
It’s 2233 in BASE 8 because 2(8³ + 8²) + 3(8 + 1) = 1179,
and 171 in BASE 31 because 31² + 7(31) + 1 = 1179

1178 School Supplies

There is so much to see at the Ópusztaszer National Heritage Park whose location marks the birthplace of the nation of Hungary. I will mention only two of its attractions in this post.

In the rotunda is a huge cyclorama painting by Arpad Feszty depicting the arrival of the Magyars over 1100 years ago. The painting in of itself is amazing, but it is also augmented with artificial landscapes in front of and all the way around the painting, giving it a 3D effect. Photography in the rotunda is forbidden, but there is no way to capture the magnificence of this work of art in a 2-dimensional photo anyway. (Neither do these few words I’ve written.)

The 15-building museum village includes a school. I was pleased to see some of the authentic school supplies from around the turn of the 20th century and before. This first one is a slate students could use not only to write mathematical calculations but also to graph equations or make bar graphs!

This abacus also made me smile.

I would encourage you to visit Ópusztaszer National Heritage Park should you get to visit Hungary, but give yourselves more than the two hours we did to enjoy all it has to offer.

Now I’ll share some information about the number 1178:

  • 1178 is a composite number.
  • Prime factorization: 1178 = 2 × 19 × 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 1178 has exactly 8 factors.
  • Factors of 1178: 1, 2, 19, 31, 38, 62, 589, 1178
  • Factor pairs: 1178 = 1 × 1178, 2 × 589, 19 × 62, or 31 × 38
  • 1178 has no square factors that allow its square root to be simplified. √1178 ≈ 34.322

1178 is a leg in a few Pythagorean triples including
600-1178-1322 calculated from  31² – 19², 2(31)(19), 31² + 19²

1178 is palindrome 212 in BASE 24 because 2(24²) + 1(24) + 2(1) = 1178

1177 and Level 3

Why are two of the clues in today’s level three puzzle in red?

You still figure out the common factor of 32 and 72, then work down the first column cell by cell filling in factors as you go, BUT you won’t be able to know what factors to use for 9 unless you look at the number 15 first. You don’t have a problem with that, do you?

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

Now I’ll tell you some facts about the number 1177:

  • 1177 is a composite number.
  • Prime factorization: 1177 = 11 × 107
  • 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 1177 has exactly 4 factors.
  • Factors of 1177: 1, 11, 107, 1177
  • Factor pairs: 1177 = 1 × 1177 or 11 × 107
  • 1177 has no square factors that allow its square root to be simplified. √1177 ≈ 34.30743

1177 is a palindrome in two bases:
It’s 414 in BASE 17 because 4(17²) + 1(17) + 4(1) = 1177
and 1E1 in BASE 28 (E is 14 base 10) because 28² + 14(28) + 1 = 1177

How to Simplify √1176

About 83% of numbers whose square root can be simplified are divisible by 4 or by 9 or by both. It’s easy to see rather quickly if a number is divisible by either of them. I want to find square roots by doing as few divisions as possible. It is easier to divide a number by 4 than it is to divide it by 2 twice. Likewise, it is easier to divide a number by 9 than it is to divide it by 3 twice.

1176 is divisible by 4 because the number formed from its last two digits, 76, is divisible by 4. So let’s do that division:
1176÷4=294.

Can we divide 294 by 4? We can quickly see that 94 is not divisible by 4, so 294 is not divisible by 4. Let’s check to see if it is divisible by nine: 2 + 9 + 4 = 15, which is not divisible by 9 but is by 3. That means that 294 is not divisible by 9, but it is divisible by 3.

Since 294 is an even number that is divisible by 3, I will save time and make fewer mistakes if I just divide it by 6 instead of by 2 and then by 3.
294÷6 = 49

49 is a perfect square so I know I’ve done enough dividing for this particular number. I’ve made a gif to illustrate the steps I’ve just explained:
How to Simplify  √1176

make science GIFs like this at MakeaGif

By doing just a little bit of arithmetic in my head, I can also tell by looking at that cake that the prime factorization of 1176 is 2³ × 3 × 7².

Some people still prefer to use factor trees to find square roots. Here are a few of its MANY possible trees.

Here’s more about the number 1176:

  • 1176 is a composite number.
  • Prime factorization: 1176 = 2 × 2 × 2 × 3 × 7 × 7, which can be written 1176 = 2³ × 3 × 7²
  • The exponents in the prime factorization are 3, 1 and 2. Adding one to each and multiplying we get (3 + 1)(1 + 1)(2 + 1) = 4 × 2 × 3 = 24. Therefore 1176 has exactly 24 factors.
  • Factors of 1176: 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 49, 56, 84, 98, 147, 168, 196, 294, 392, 588, 1176
  • Factor pairs: 1176 = 1 × 1176, 2 × 588, 3 × 392, 4 × 294, 6 × 196, 7 × 168, 8 × 147, 12 × 98, 14 × 84, 21 × 56, 24 × 49 or 28 × 42
  • Taking the factor pair with the largest square number factor, we get √1176 = (√196)(√6) = 14√6 ≈ 34.29286

1176 is the sum of the first 48 numbers so we say it the 48th triangular number. We know it is the sum of the first 48 numbers because (48 × 49)/2 = 1176

1176 looks interesting in a few other bases:
It’s 3300 in BASE 7 because 3(7³ + 7²) = 3(392) = 1176,
6C6 in BASE 13 (C is 12 base 10) because 6(13²) + 12(13) + 6(1) = 1176,
600 in BASE 14 because 6(14²) = 1176,
1M1 in BASE 25 (M is 22 base 10) because 25² + 22(25) + 1 = 1176
and it looks like one of its factors, 196, in BASE 30
because 1(30²) + 9(30) + 6(1) = 1176