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

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1175 and Level 2

This level 2 puzzle will be quite useful in helping you recall all the multiplication facts. How many factors can you fill in on this puzzle without looking at a regular multiplication table? I congratulate you on all the ones you know.

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

Here are some facts about the number 1175:

  • 1175 is a composite number.
  • Prime factorization: 1175 = 5 × 5 × 47, which can be written 1175 = 5² × 47
  • 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 1175 has exactly 6 factors.
  • Factors of 1175: 1, 5, 25, 47, 235, 1175
  • Factor pairs: 1175 = 1 × 1175, 5 × 235, or 25 × 47
  • Taking the factor pair with the largest square number factor, we get √1175 = (√25)(√47) = 5√47 ≈ 34.27827

1175 is the hypotenuse of two Pythagorean triples:
705-940-1175 which is (3-4-5) times 235
329-1128-1175 which is (7-24-25) times 47

1175 is a palindrome in these other bases:
It’s 979 in BASE 11 because 9(11²) + 7(11) + 9(1) = 1175,
535 in BASE 15 because 5(15²) + 3(15) + 5(1) = 1175,
and 252 in BASE 23 because 2(23²) + 5(23) + 2(1) = 1175

1174 and Level 1

I’ve given you just nine clues in this puzzle, but that’s enough to find all the factors AND complete the entire table. I’m serious. I really have given you sufficient information to find the one and only solution to this puzzle!

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

Now I’ll share some information about the number 1174:

  • 1174 is a composite number.
  • Prime factorization: 1174 = 2 × 587
  • 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 1174 has exactly 4 factors.
  • Factors of 1174: 1, 2, 587, 1174
  • Factor pairs: 1174 = 1 × 1174 or 2 × 587
  • 1174 has no square factors that allow its square root to be simplified. √1174 ≈ 34.26368

1174 is also the sum of the sixteen prime numbers from 41 to 107.

 

1173 Challenge Puzzle

Getting started on this Challenge Puzzle will take some thinking, but solving it is worth all the effort. Remember use logic, not guess and check, and you will eventually be successful!

Print the puzzles or type the solution in this excel file: 12 factors 1161-1173

Here’s some information about the number 1173:

  • 1173 is a composite number.
  • Prime factorization: 1173 = 3 × 17 × 23
  • 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 1173 has exactly 8 factors.
  • Factors of 1173: 1, 3, 17, 23, 51, 69, 391, 1173
  • Factor pairs: 1173 = 1 × 1173, 3 × 391, 17 × 69, or 23 × 51
  • 1173 has no square factors that allow its square root to be simplified. √1173 ≈ 34.24909

1173 is the hypotenuse of a Pythagorean triple:
552-1035-1173 which is (8-15-17) times 69

1173 is palindrome 3B3 in BASE 18 (B is 11 base 10)
because 3(18²) + 11(18) + 3(1) = 1173

1172 Mystery Puzzle

There are thirteen clues in this Mystery Level Find the Factors 1 – 12 puzzle. Will those thirteen clues bring you good luck or bad? The logic needed to solve the puzzle may be a bit complicated, but if you stick with it, you will figure it out. Good luck to you!

Print the puzzles or type the solution in this excel file: 12 factors 1161-1173

Let me share some facts about the number 1172:

  • 1172 is a composite number.
  • Prime factorization: 1172 = 2 × 2 × 293, which can be written 1172 = 2² × 293
  • 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 1172 has exactly 6 factors.
  • Factors of 1172: 1, 2, 4, 293, 586, 1172
  • Factor pairs: 1172 = 1 × 1172, 2 × 586, or 4 × 293
  • Taking the factor pair with the largest square number factor, we get √1172 = (√4)(√293) = 2√293 ≈ 34.23449

1172 is the sum of six consecutive prime numbers:
181 + 191 + 193 + 197 + 199 + 211 = 1172

34² + 4² = 1172

1172 is the hypotenuse of a Pythagorean triple:
272-1140-1172 calculated from 2(34)(4), 34² – 4², 34² + 4²
It is also 4 times (68-285-293)

1172 is a palindrome in a couple of bases:
It’s 818 in BASE 12 because 8(12²) + 1(12) + 8(1) = 1172,
and 494 in BASE 16 4(16²) + 9(16) + 4(1) = 1172

1171 The Best Team in the Best Conference

The NCAA college football season has not had a single game, yet you can find out which team is in first place through twenty-fifth place now or anytime during the season by looking here. How are these football standings determined? By FIFTEEN people voting. Sure, it’s only one of several polls, but the four teams who play for the national championship are determined by a computer that uses polls like that one. Can you believe that there are people who find that rather unsatisfying? Your team could finish the season with the exact same record as one of those four teams but not be allowed to compete for the championship.

What do college football teams have to play for then? Almost every team is in a conference. They can play hoping to win their conference. Those teams who have a winning record can also be selected to play in one of 38 bowl games in December or early January. Winning a bowl game allows a team to finish the season with a win and is an honor to the school. Other than that, 35 of those bowl games mean absolutely nothing.

Perhaps this is a bit simplistic, but why can’t each conference send their best teams to play in bowl games against teams from a different conference. The conference that wins the most bowl games would be deemed the best conference.  The team that won that conference’s championship would be the best team in the best conference and the national champion. Every bowl game would then be important. Each eligible team would still only have to play one bowl game. More people would watch EVERY bowl game which would cause them all to make more money. The sports stations would also make more money as they keep their viewers updated with the win/loss records for every conference week after week.

Of all the things that are happening in the world today, this issue is far from being the most important, but thinking about it, like sports or this football-shaped mystery level puzzle, is a nice diversion.

Print the puzzles or type the solution in this excel file: 12 factors 1161-1173

Here’s the same puzzle but without all the color.

Now I’ll write a few things about the number 1171:

  • 1171 is a prime number.
  • Prime factorization: 1171 is prime.
  • The exponent of prime number 1171 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1171 has exactly 2 factors.
  • Factors of 1171: 1, 1171
  • Factor pairs: 1171 = 1 × 1171
  • 1171 has no square factors that allow its square root to be simplified. √1171 ≈ 34.21988

How do we know that 1171 is a prime number? If 1171 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1171 ≈ 34.2. Since 1171 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1171 is a prime number.

1171 is the sum of seven consecutive prime numbers:
151 + 157 + 163 + 167 + 173 + 179 + 181 = 1171

1171 is a palindrome in three bases:
It’s 14141 in BASE 5 because 5⁴ + 4(5³) + 5² + 4(5) + 1 = 1171,
1J1 in BASE 26 (J is 19 base 10) because 26² + 19(26) + 1 = 1171,
and 191 in BASE 30 because 30² + 9(30) + 1 = 1171

1170 Factor Trees

1170 is one of those numbers with twenty-four factors. Why does it have so many? Because of its prime factorization. You can find a number’s prime factorization by making a factor tree. Here are eleven different factor trees for 1170. Each one of them produces the same prime factorization: 1170 = 2 × 3² × 5 × 13

  • 1170 is a composite number.
  • Prime factorization: 1170 = 2 × 3 × 3 × 5 × 13, which can be written 1170 = 2 × 3² × 5 × 13
  • The exponents in the prime factorization are 1, 2, 1, and 1. Adding one to each and multiplying we get (1 + 1)(2 + 1)(1 + 1)(1 + 1) = 2 × 3 × 2 × 2 = 24. Therefore 1170 has exactly 24 factors.
  • Factors of 1170: 1, 2, 3, 5, 6, 9, 10, 13, 15, 18, 26, 30, 39, 45, 65, 78, 90, 117, 130, 195, 234, 390, 585, 1170
  • Factor pairs: 1170 = 1 × 1170, 2 × 585, 3 × 390, 5 × 234, 6 × 195, 9 × 130, 10 × 117, 13 × 90, 15 × 78, 18 × 65, 26 × 45, or 30 × 39
  • Taking the factor pair with the largest square number factor, we get √1170 = (√9)(√130) = 3√130 ≈ 34.20526

33² +  9² = 1170
27² +  21² = 1170

1170 is also the hypotenuse of FOUR Pythagorean triples:
288-1134-1170 calculated from 27² –  21², 2(27)(21), 27² +  21²
450-1080-1170 which is (5-12-13) times 90
594-1008-1170 calculated from 2(33)( 9), 33² –  9², 33² +  9²
702-936-1170 which is (3-4-5) times 234

1170 is 102102 in BASE 4 because 4⁵ + 2(4³) +4² + 2(1) = 1170,
and it’s repdigit 2222 in BASE 8 because 2(8³ + 8² + 8¹ + 8⁰) = 2(585) = 1170

1169 and Level 6

The twelve clues in this puzzle make an attractive puzzle for you to solve. What factors go with those clues? Can you find the logic needed to figure this one out?

Print the puzzles or type the solution in this excel file: 12 factors 1161-1173

Now I’ll share what I’ve learned about the number 1169:

  • 1169 is a composite number.
  • Prime factorization: 1169 = 7 × 167
  • 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 1169 has exactly 4 factors.
  • Factors of 1169: 1, 7, 167, 1169
  • Factor pairs: 1169 = 1 × 1169 or 7 × 167
  • 1169 has no square factors that allow its square root to be simplified. √1169 ≈ 34.19064

1169 is the sum of consecutive prime numbers two different ways:
227+ 229 + 233+ 239+ 241 = 1169
383+ 389 + 397 = 1169

1169 is palindrome 5225 in BASE 6 because 5(6³) + 2(6²) + 2(6) + 5(1) = 1169

1168 Sosto Museum Village School Room

During our recent visit to Nyíregyháza, Hungary we visited the Sosto Museum Village. One of my favorite places there was this school room.

The room was roped off so I had to settle for this shot from the doorway. Let me tell you what I see in this picture.

On the right of the picture is an abacus. As a lover of mathematics, I have to love that there was an abacus in the classroom.

At the top of the page near the center is a map of Szabolcs and Ung Counties. Ung county was where my husband’s maternal grandfather was born and was only about 75 km from this museum village. I like to think that his grandfather’s classroom might have been just like this one.

I love the ceiling with its wooden beams as well as the desks and other wood furnishings in the room. My husband’s paternal grandfather was a cabinet maker. The second cousins we met in Romania informed us that this grandfather made the desks at their school. Even though that school was far away from this museum village, I imagine that the desks he made looked much like these.

In a different classroom, we found this mathematics book. We could walk up and look at it quite easily, but we couldn’t turn any of the pages because it was behind glass. I apologize for the glare from the glass. They don’t make arithmetic books like this anymore!

One of the classrooms had this guide for reading and writing the alphabet.

Some other pictures of the museum village can be found here. I took other pictures, but this is enough for this post. I recommend going to Sosto Museum should you ever travel to Hungary.

Now I’ll write a little about the number 1168:

  • 1168 is a composite number.
  • Prime factorization: 1168 = 2 × 2 × 2 × 2 × 73, which can be written 1168 = 2⁴ × 73
  • The exponents in the prime factorization are 4 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1) = 5 × 2 = 10. Therefore 1168 has exactly 10 factors.
  • Factors of 1168: 1, 2, 4, 8, 16, 73, 146, 292, 584, 1168
  • Factor pairs: 1168 = 1 × 1168, 2 × 584, 4 × 292, 8 × 146, or 16 × 73
  • Taking the factor pair with the largest square number factor, we get √1168 = (√16)(√73) = 4√73 ≈ 34.17601

1168 is the hypotenuse of a Pythagorean triple:
768-880-1168 which is 16 times (48-55-73)

1168 is palindrome 292 in BASE 22 because 2(22²) + 9(22) + 2(1) = 1168

1167 and Level 5

Will some of the tricky clues in this level 5 puzzle fool you? They won’t if you only write factors of which you are 100% sure. Always use logic. Never guess and check.

Print the puzzles or type the solution in this excel file: 12 factors 1161-1173

Now I’ll write a little bit about the number 1167:

  • 1167 is a composite number.
  • Prime factorization: 1167 = 3 × 389
  • 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 1167 has exactly 4 factors.
  • Factors of 1167: 1, 3, 389, 1167
  • Factor pairs: 1167 = 1 × 1167 or 3 × 389
  • 1167 has no square factors that allow its square root to be simplified. √1167 ≈ 34.16138

1167 is the hypotenuse of a Pythagorean triple:
567-1020-1167 which is 3 times (189-340-389)

1167 is palindrome 5D5 in BASE 14 (D is 13 base 10)
because 5(14²) + 13(14) + 5(1) = 1167