1123 and Level 2

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All the clues in one of the rows of this Level 2 puzzle are prime numbers. The only common factor they have is 1. That fact will get you started with this puzzle which I am sure you can complete if you just give it a try!

Print the puzzles or type the solution in this excel file: 10-factors-1121-1133

1123 is also a prime number. Here are some facts about it.

  • 1123 is a prime number.
  • Prime factorization: 1123 is prime.
  • The exponent of prime number 1123 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1123 has exactly 2 factors.
  • Factors of 1123: 1, 1123
  • Factor pairs: 1123 = 1 × 1123
  • 1123 has no square factors that allow its square root to be simplified. √1123 ≈ 33.51119

How do we know that 1123 is a prime number? If 1123 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1123 ≈ 33.5. Since 1123 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1123 is a prime number.

OEIS.org reminds us that 1, 1, 2, 3 are the first four numbers in the Fibonacci sequence.

1123 is the sum of five consecutive prime numbers:
211 + 223 + 227 + 229 + 233 = 1123

1123 is palindrome 797 in BASE 12 because 7(12²) + 9(12) + 7(1) = 1123, and
it’s repdigit 111 in BASE 33 because 33² + 33 + 1 = 1123

1122 Factor Trees

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We can easily tell that 1122 is divisible by 2 because it is even.
It is divisible by 3 because 1 + 2 = 3, no matter how many times you do it.
11 and 22 are both divisible by 11, so obviously, 1122 is as well.
1122 is also divisible by 17, but there isn’t an easy trick to use to find that out without just dividing it.

There are 24 ways to arrange the four factors of 1122. Each of those 24 ways can be the bottom four numbers of a factor tree (See the first three trees below.) or the numbers running down the side of a factor tree (See the last four trees below.).

1122 makes some very well proportioned factor trees!

  • 1122 is a composite number.
  • Prime factorization: 1122 = 2 × 3 × 11 × 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 = 16. Therefore 1122 has exactly 16 factors.
  • Factors of 1122: 1, 2, 3, 6, 11, 17, 22, 33, 34, 51, 66, 102, 187, 374, 561, 1122
  • Factor pairs: 1122 = 1 × 1122, 2 × 561, 3 × 374, 6 × 187, 11 × 102, 17 × 66, 22 × 51, or 33 × 34
  • 1122 has no square factors that allow its square root to be simplified. √1122 ≈ 33.49627

1122 is the product of two consecutive number: 33 × 34 = 1122
That makes 1122 the sum of the first 33 even numbers:
2 + 4 + 6 + 8 + . . . + 62 + 64 + 66 = 1122

1122 is the hypotenuse of a Pythagorean triple:
528-990-1122 which is (8-15-17) times 66

1122 is palindrome 2G2 in BASE 20 (G is 16 base 10)
because 2(20²) + 16(20) + 2(1) = 1122

1121 and Level 1

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If you’ve learned how to multiply and divide, then you can solve this puzzle. Just write the numbers from 1 to 10 in both the first column and the top row so that the clues and those factors make a multiplication table. You can definitely do this one!

Print the puzzles or type the solution in this excel file: 10-factors-1121-1133

Here are a few facts about the number 1121:

  • 1121 is a composite number.
  • Prime factorization: 1121 = 19 × 59
  • 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 1121 has exactly 4 factors.
  • Factors of 1121: 1, 19, 59, 1121
  • Factor pairs: 1121 = 1 × 1121 or 19 × 59
  • 1121 has no square factors that allow its square root to be simplified. √1121 ≈ 33.48134

1121 is a palindrome in two other bases:
It’s 1C1 in BASE 28 (C is 12 base 10) because 28² + 12(28) + 1 = 1121,
and it’s 131 in BASE 32 because 32² + 3(32) + 1 = 1121

1120 There’s Lots of Shade under These Factor Trees!

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It’s hot this summer, but you can rest under the shade of some of 1120’s factor trees. Whether you want a tall tree or a wide one, these are only a few of the MANY possible ones you can choose. Every one of them gives you the same prime factors which are shown in pink here.

Here are some things I’ve learned about the number 1120:

  • 1120 is a composite number.
  • Prime factorization: 1120 = 2 × 2 × 2 × 2 × 2 × 5 × 7, which can be written 1120 = 2⁵ × 5 × 7
  • The exponents in the prime factorization are 5, 1, and 1. Adding one to each and multiplying we get (5 + 1)(1 + 1)(1 + 1) = 7 × 2 × 2 = 28. Therefore 1120 has exactly 28 factors.
  • Factors of 1120: 1, 2, 4, 5, 7, 8, 10, 14, 16, 20, 28, 32, 35, 40, 56, 70, 80, 112, 140, 160, 224, 280, 560, 1120
  • Factor pairs: 1120 = 1 × 1120, 2 × 560, 4 × 280, 5 × 224, 7 × 160, 8 × 140, 10 × 112, 14 × 80, 16 × 70, 20 × 56, 28 × 40, or 32 × 35
  • Taking the factor pair with the largest square number factor, we get √1120 = (√16)(√70) = 4√70 ≈ 33.4664

1120 is the sum of the twenty prime numbers from 17 to 101.
It is also the sum of these two consecutive primes:
557 + 563 = 1120

1120 is the hypotenuse of exactly one Pythagorean triple:
672-896-1120 which is (3-4-5) times 224

I like the way 1120 looks in these other bases:
It’s 1112111 in BASE 3 because 3⁶ + 3⁵ + 3⁴ + 2(3³) + 3² + 3  + 1 = 1120,
929 in BASE 11 because 9(11²) + 2(11) + 9(1) = 1120,
WW in BASE 34 (W is 32 base 10) because 32(34) + 32(1) = 32(35) = 1120,
and W0 in BASE 35 because 32(35) = 1120

1119 Mystery

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Will this Mystery Level Puzzle be easy or difficult? That is part of the mystery! You might find the logic needed to solve it a mystery, too, but I assure you it can be solved without guessing and checking. Can you do it?

Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

Here are some facts about the number 1119:

  • 1119 is a composite number.
  • Prime factorization: 1119 = 3 × 373
  • 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 1119 has exactly 4 factors.
  • Factors of 1119: 1, 3, 373, 1119
  • Factor pairs: 1119 = 1 × 1119 or 3 × 373
  • 1119 has no square factors that allow its square root to be simplified. √1119 ≈ 33.45146

1119 is the sum of three consecutive prime numbers, one of which is its prime factor:
367 + 373 + 379 = 1119

1119 is the hypotenuse of a Pythagorean triple:
756-825-1119 which is 3 times (252-275-373)

1119 is a palindrome in two other bases:
It’s 383 in BASE 18 because 3(18²) + 8(18) + 3(1) = 1119, and
1H1 in BASE 26 (H is 17 base 10) because 1(26²) + 17(26) + 1(1) = 1119

1118 and Level 6

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Each Level 6 puzzle has different logic needed to solve it. For this one, you will ask yourself over and over again, “Where is the only place ___ can go in the first column (or the top row)?” Just stick with it until all the numbers from 1 to 12 are in both of those places.

Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

Now here’s a little about the number 1118:

  • 1118 is a composite number.
  • Prime factorization: 1118 = 2 × 13 × 43
  • 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 1118 has exactly 8 factors.
  • Factors of 1118: 1, 2, 13, 26, 43, 86, 559, 1118
  • Factor pairs: 1118 = 1 × 1118, 2 × 559, 13 × 86, or 26 × 43
  • 1118 has no square factors that allow its square root to be simplified. √1118 ≈ 33.43651

1118 is the hypotenuse of a Pythagorean triple:
430-1032-1118 which is (5-12-13) times 86

1118 is also a leg in one of this year’s Pythagorean triples!
1118-1680-2018 calculated from 2(43)(13), 43² – 13², 43² + 13²

1117 and Level 5

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Can you solve this Level 5 puzzle or will it make you feel like toast? Seriously, I think it’s a little easier than the previous puzzle, so give it a try. You might just become the toast of the town!

Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

Now I’ll share some information about the number 1117:

  • 1117 is a prime number.
  • Prime factorization: 1117 is prime.
  • The exponent of prime number 1117 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1117 has exactly 2 factors.
  • Factors of 1117: 1, 1117
  • Factor pairs: 1117 = 1 × 1117
  • 1117 has no square factors that allow its square root to be simplified. √1117 ≈ 33.42155

How do we know that 1117 is a prime number? If 1117 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1117 ≈ 33.4. Since 1117 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1117 is a prime number.

26² + 21² = 1117

1117 is the hypotenuse of a Pythagorean triple:
235-1092-1117 calculated from 26² – 21², 2(26)(21), 26² + 21²

Here’s another way we know that 1117 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 26² + 21² = 1117 with 26 and 21 having no common prime factors, 1117 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1117 ≈ 33.4. Since 1117 is not divisible by 5, 13, 17, or 29, we know that 1117 is a prime number.

1117 is palindrome 151 in BASE 31 because 1(31²) + 5(31) + 1(1) = 1117

1116 Can I Trick You With This Puzzle?

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This is only a Level 4 puzzle, but can I trick you into making some assumptions about the puzzle’s factors? I bet I will trick a least a few people. Will you be one of them? If you don’t write a factor unless you are 100% sure it goes where you are putting it, I shouldn’t be able to trick you. Good luck with this one!

Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

Now I’ll share some facts about the number 1116:

  • 1116 is a composite number.
  • Prime factorization: 1116 = 2 × 2 × 3 × 3 × 31, which can be written 1116 = 2² × 3² × 31
  • The exponents in the prime factorization are 2, 2 and 1. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 2 = 18. Therefore 1116 has exactly 18 factors.
  • Factors of 1116: 1, 2, 3, 4, 6, 9, 12, 18, 31, 36, 62, 93, 124, 186, 279, 372, 558, 1116
  • Factor pairs: 1116 = 1 × 1116, 2 × 558, 3 × 372, 4 × 279, 6 × 186, 9 × 124, 12 × 93, 18 × 62 or 31 × 36
  • Taking the factor pair with the largest square number factor, we get √1116 = (√36)(√31) = 6√31 ≈ 33.40659

1116 looks interesting when it is represented in these other bases:
It’s 13431 in BASE 5 because 1(5⁴) + 3(5³) + 4(5²) + 3(5) + 1(1) = 1116,
VV in BASE 35 (V is 31 base 10) because 31(35) + 31(1) = 31(36) = 1116, and
V0 in BASE 36 because 31(36) = 1116

 

1115 and Level 3

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You will have to know the 11 and 12 times tables to solve this Level 3 Find the Factors 1-12 puzzle, but I’m sure you can do it! Stick with it, and don’t give up! Start with the clues at the top of the puzzle and work down row by row until it’s completed.

Print the puzzles or type the solution in this excel file: 12 factors 1111-1119

1115 is not a clue just the puzzle number. Here are some facts about the number 1115:

  • 1115 is a composite number.
  • Prime factorization: 1115 = 5 × 223
  • 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 1115 has exactly 4 factors.
  • Factors of 1115: 1, 5, 223, 1115
  • Factor pairs: 1115 = 1 × 1115 or 5 × 223
  • 1115 has no square factors that allow its square root to be simplified. √1115 ≈ 33.39162

1115 is the sum of nine consecutive prime numbers:
103 + 107 + 109 + 113 + 127 + 131 + 137 + 139 + 149 = 1115

1115 is the hypotenuse of a Pythagorean triple:
669-892-1115 which is (3-4-5) times 223

1115 is palindrome 2B2 in BASE 21 (B is 11 base 10)
because 2(21²) + 11(21) + 2(1) = 1115

 

1114 My Rant about the State’s Emissions Test

/ ivasallay / 2 Comments

The valley I live in is often the worst place in the country for air pollution. Of course, when we register our cars we have to prove that the car passes the state’s emissions test. Naturally, it sounds like a good idea. In the past, our cars have always passed their emissions tests, but this year I want to rant about how POLLUTING the emissions test can be.

Two and half weeks before our car registration was due, I took our Honda Civic in for the emissions test. It didn’t pass. It didn’t fail either. I was simply told that the car was not ready for the test. We hadn’t driven it enough. I listened in disbelief. “Drive it around some more and come back,” the service person told me. Something on the car was reset the last time I took the car in for a repair or maintenance, so now the car needs more miles on it to be ready for the test.

We drove the car around and had it checked two more times before we left for Europe, but the car still wasn’t ready. Every place we went the weekend before we left, we took that car. Then I asked a neighbor to drive it some the two weeks we were gone. She didn’t drive it much, but we’ve put about 100 miles on the car since we’ve returned. I took it in to get checked again today. It still isn’t ready.

It’s hot outside. There hasn’t been much rain. There is a wildfire close enough that we can smell the smoke. The mountains in the distance look faded. This is not the time for us to needlessly add more pollutants to the air as we drive it around just to make the car ready for the test.  All cars pollute the air. Everyone in the area is encouraged to carpool or combine trips to try to keep the pollution lower. We, on the other hand, need to drive our car aimlessly around just to put more miles on it so hopefully, the car will be ready for the emissions test. I have a very difficult time doing that, so instead of driving it around, I’m sitting at my computer ranting. The car needs to pass the test by Tuesday or we will have to pay for another test. This whole thing is a waste of time, gasoline, and money. Meanwhile, we couldn’t register our car, so potentially we could get in trouble for driving a car that isn’t registered, too.

Writing about the number 1114 should be a much more pleasant subject than car emissions!

  • 1114 is a composite number.
  • Prime factorization: 1114 = 2 × 557
  • 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 1114 has exactly 4 factors.
  • Factors of 1114: 1, 2, 557, 1114
  • Factor pairs: 1114 = 1 × 1114 or 2 × 557
  • 1114 has no square factors that allow its square root to be simplified. √1114 ≈ 33.37664

33² + 5² = 1114

1114 is the hypotenuse of a Pythagorean triple:
330-1064-1114 calculated from 2(33)(5), 33² – 5², 33² + 5²

1114 is the sum of 10 consecutive primes and 6 consecutive primes:
89 + 97 + 101 + 103 + 107 + 109 + 113 + 127 + 131 + 137 = 1114,
and 173 + 179 + 181 + 191 + 193 + 197 = 1114

1114 is palindrome 4E4 in BASE 15 (E is 14 base 10)
because 4(15²) + 14(15) + 4(1) = 1114