A Multiplication Based Logic Puzzle

Posts tagged ‘prime’

1033 and Level 3

To solve a level 3 puzzle begin with 80, the clue at the very top of the puzzle. Clue 48 goes with it. What are the factor pairs of those numbers in which both factors are between 1 and 12 inclusive? 80 can be 8×10, and 48 can be 4×12 or 6×8. What is the only number that listed for both 80 and 48? Put that number in the top row over the 80. Put the corresponding factors where they go starting at the top of the first column.

Work down that first column cell by cell finding factors and writing them as you go. Three of the factors have been highlighted because you have to at least look at the 55 and the 5 to deal with the 20 in the puzzle. Have fun!

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

Here are a few facts about the number 1033:

It is a twin prime with 1031.

32² + 3² = 1033, so it is the hypotenuse of a Pythagorean triple:
192-1015-1033 calculated from 2(32)(3), 32² – 3², 32² + 3²

1033 is a palindrome in two other bases:
It’s 616 in BASE 13 because 6(13²) + 1(13) + 6(1) = 1033
1J1 in BASE 24 (J is 19 base 10) because 24² + 19(24) + 1 = 1033

8¹ + 8º + 8³ + 8³ = 1033 Thanks to Stetson.edu for that fun fact!

  • 1033 is a prime number.
  • Prime factorization: 1033 is prime.
  • The exponent of prime number 1033 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1033 has exactly 2 factors.
  • Factors of 1033: 1, 1033
  • Factor pairs: 1033 = 1 × 1033
  • 1033 has no square factors that allow its square root to be simplified. √1033 ≈ 32.1403

How do we know that 1033 is a prime number? If 1033 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1033 ≈ 32.1. Since 1033 cannot be divided evenly by 2, 3, 5, 7, 3, 13, 17, 19, 23, 29 or 31, we know that 1033 is a prime number.

Here’s another way we know that 1033 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 32² + 3² = 1033 with 32 and 3 having no common prime factors, 1033 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1033 ≈ 32.1. Since 1033 is not divisible by 5, 13, 17, or 29, we know that 1033 is a prime number.

 

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1031 Prepare for World Maths Day 2018

You might think that a day lasts 24 hours, but strategic use of the international date line can actually make a single day last 48 hours!

How will you spend the 48 hour day that will be 7 March 2018?

Colleen Young encourages you and your class to register for and participate in World Maths Day 2018 held that day. She shares the necessary links as well as several tips on how to prepare.

One way to prepare now is playing multiplication games like the level 1 puzzle below. The puzzle is just a multiplication table but the factors are missing and only a few of the products are given, and they aren’t in the order you would normally expect. Can you figure out where the factors from 1 to 12 belong in both the first column and the top row of the puzzle?

Print the puzzles or type the solution in this excel file: 12 factors 1028-1034

If this puzzle is too easy for you, Then it is time to move on to a level 2 or higher puzzle. You can find one in the link above and plenty others here at findthefactors.com.

Now I’d like to tell you some things that I’ve learned about the number 1031:

1031 and 1033 are twin primes.

1031 is a palindrome in a couple of bases:
It’s 858 in BASE 11 because 8(121) + 5(11) + 8(1) = 1031 and
it’s 272 in BASE 21 because 2(441) + 7(21) + 2(1) = 1031

  • 1031 is a prime number.
  • Prime factorization: 1031 is prime.
  • The exponent of prime number 1031 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1031 has exactly 2 factors.
  • Factors of 1031: 1, 1031
  • Factor pairs: 1031 = 1 × 1031
  • 1031 has no square factors that allow its square root to be simplified. √1031 ≈ 32.10919

How do we know that 1031 is a prime number? If 1031 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1031 ≈ 32.1. Since 1031 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1031 is a prime number.

1021 Mystery Level

Here is the second puzzle in a week’s worth of mystery level puzzles. Will it be very difficult or not so bad? That’s the mystery. You’ll have to try it to know for sure. You only need to use logic and your knowledge of the multiplication table to solve it. I wish you luck!

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

That puzzle may have been a mystery, but the number 1021 isn’t much of a mystery at all. It is the second number of twin primes, 1019 and 1021.

Since it is a prime number, and it has a remainder of one when it is divided by 4, it can be written as the sum of two squares:
30² + 11² = 1021.

Since it can be written as the sum of two squares, it is the hypotenuse of a Pythagorean triple:
660-779-1021 calculated from 2(30)(11), 30² – 11², 30² + 11²

It is also palindrome 141 in BASE 30 because 1(30²) + 4(30) + 1(1) = 1021

  • 1021 is a prime number.
  • Prime factorization: 1021 is prime.
  • The exponent of prime number 1021 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1021 has exactly 2 factors.
  • Factors of 1021: 1, 1021
  • Factor pairs: 1021 = 1 × 1021
  • 1021 has no square factors that allow its square root to be simplified. √1021 ≈ 31.95309

How do we know that 1021 is a prime number? If 1021 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1021 ≈ 31.95. Since 1021 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1021 is a prime number.

Here’s another way we know that 1021 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 30² + 11² = 1021 with 30 and 11 having no common prime factors, 1021 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1021 ≈ 31.95. Since 1021 is not divisible by 5, 13, 17, or 29, we know that 1021 is a prime number.

1019 An Easier Find the Factors Challenge Puzzle

I’ve recently posted some more challenging puzzles that I’ve named Find the Factors 1 – 10 Challenge, and they definitely are a more challenging puzzle than one of my more traditional level 6 puzzles. As of today, no one has informed me that they have been able to solve either puzzle number 1000 or 1010.

Two years ago I made perhaps my most challenging level 6 puzzle, a 16 × 16 puzzle to commemorate Steve Morris’s birthday. Steve Morris was the very first person to type a comment on my blog, and I have appreciated his encouragement over the years. Steve has solved many kinds of puzzles in his life including some of the toughest I have made, but the puzzle I made for that birthday was no picnic for even him to complete.

This year I’ve made him a challenging puzzle, but it is still a little easier than the other two challenge puzzles I’ve made. If you’ve tried either of those other puzzles without success, still give this one a try. Good luck to you all, and Happy Birthday to Steve Morris! I saved this post number (1019) for you because it uses your birthdate numbers, howbeit out of order.

Print the puzzles or type the solution in this excel file: 10-factors-1019-1027

This is my 1019th post. Here are a few facts about the number 1019.

Prime number 1019 is the sum of the 19 prime numbers from 17 to 97.

  • 1019 is a prime number.
  • Prime factorization: 1019 is prime.
  • The exponent of prime number 1019 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1019 has exactly 2 factors.
  • Factors of 1019: 1, 1019
  • Factor pairs: 1019 = 1 × 1019
  • 1019 has no square factors that allow its square root to be simplified. √1019 ≈ 31.921779

How do we know that 1019 is a prime number? If 1019 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1019 ≈ 31.9. Since 1019 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1019 is a prime number.

997 and Level 3

Can you fill in all the cells of this 12 × 12 mixed up multiplication table if all you are given are the clues given in this puzzle? I promise you it can be done. Start with the two clues at the top of the puzzle and work down clue by clue until you have found all the factors. Afterwards, filling in the rest of the multiplication table will be a breeze.

Print the puzzles or type the solution in this excel file: 12 factors 993-1001

Here are a few facts about 997, the largest three-digit prime number:

31² + 6²  = 997
That means 997 is the hypotenuse of a primitive Pythagorean triple:
372-925-997 calculated from 2(31)(6), 31² – 6², 31² + 6²

  • 997 is a prime number.
  • Prime factorization: 997 is prime.
  • The exponent of prime number 997 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 997 has exactly 2 factors.
  • Factors of 997: 1, 997
  • Factor pairs: 997 = 1 × 997
  • 997 has no square factors that allow its square root to be simplified. √997 ≈ 31.5753

How do we know that 997 is a prime number? If 997 were not a prime number, then it would be divisible by at least one prime number less than or equal to √997 ≈ 31.3. Since 997 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 997 is a prime number.

Here’s another way we know that 997 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 31² + 6² = 997 with 31 and 6 having no common prime factors, 997 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √997 ≈ 31.6. Since 997 is not divisible by 5, 13, 17, or 29, we know that 997 is a prime number.

991 Carry Your Own Weather

This snowflake puzzle isn’t for beginners, but making snowflakes goes very well with the idea of carrying your own weather.

Print the puzzles or type the solution in this excel file: 10-factors-986-992

I like when I can sneak a little bit of mathematics into a completely unrelated lesson. This lesson is about being proactive and is called “Carry Your Own Weather”.

Carry Your Own Weather (Be Proactive) discussion:

  1. If you could choose the weather, what kind of weather would you choose? (Sunny weather and snowy weather seem to be chosen the most.)
  2. If the weather outside was ALWAYS sunny, would you like that? Would that be a good thing? Would you appreciate the sunny days if every day was sunny? (Variety is good. Without rain and snow, how could food grow?)
  3. Does your mood depend on the weather?
  4. How do you feel when it’s sunny outside?
  5. How do you feel when it’s gloomy outside?
  6. The author of the 7 Habits of Highly Effective People, Stephen R. Covey, talked about carrying your own weather. What do you think that might mean? (Decide for yourself how you will feel. Don’t let the outside weather or other people make that decision for you.)
  7. Let’s listen as Stephen R. Covey talks about Carrying Your Own Weather:

7Habits – CARRY YOUR OWN WEATHER VIDEO.flv.

Have you ever made a snowflake before? Did you know that you can decide how the snowflake will look before you make a single cut? Choosing how the snowflake will look ahead of time is like deciding what kind of weather you will carry with you. You will not leave it up to chance. You will begin with the end in mind. You will decide ahead of time how your snowflake will look. In real life, a snowflake has 6 sides, but you can choose to make your snowflake have four sides, six sides or eight sides. Several layers of a napkin are much easier to cut than the same number of layers of regular paper so you will use white paper napkins to make your snowflakes. Afterwards, you can glue your snowflake onto a sheet of dark blue construction paper. Then you can use a white crayon to sign your name and decorate the dark blue construction paper around your snowflake.

First, you need to fold your napkin. Folding into eighths is the easiest. Just find the corner where all the folds in the napkin already meet and fold that corner again, thus making a 45° angle. Fold that corner in half again and you’ll get the 22.5° angle that you see at the bottom of the napkin in the picture below. The other napkin is folded into sixths and then into twelfths. If you don’t cut off the uneven edges at the top of those napkins, your snowflake will only have four sides, but if you do cut off the uneven edges, you will get a six-sided or an eight-sided snowflake. (Depending on that cut, you might also get a twelve-sided or sixteen-sided snowflake. They’ll look great, too!)

Making a perfect six-sided snowflake is a little more difficult than an eight-sided one. I recommend reading this post from Paula Beardell Krieg for complete instructions on six-sided snowflake cutting: ‘Tis the Season to Make Paper Snowflakes (She is the one who told me about using easy-to-cut paper napkins for the snowflakes, too.)

I found these three triangles helpful in making snowflakes with perfect 60° angles.

 

Place the center of the folded napkin at the bottom center of triangles. You can easily see through the napkin to see where the folds need to go.

When each side of the napkin has been folded up, it will look have a 60° angle at the bottom. The red line shows where to cut the top off the napkin to get a perfect hexagon folded into an equilateral triangle.

Fold the napkin in half again so that bottom angle becomes a 30° angle before making your decorative cuts. These next instructions tell how to make those cuts to get the exact snowflakes that you want. These tips were made for six-sided snowflakes, but you can also apply the tips to eight-sided snowflakes:

Paper Snowflake Cutting Tips

Please be aware that these snowflakes are delicate. They can rip easily. I suggest you mount them on sturdy paper as soon as possible.

After you finish making your snowflakes, I’m sure you will enjoy a story that I like very much, even though you have heard it many times before. The main character felt sorry for himself because he was bullied and nobody thought he was good at anything. When he decided to find a way to be helpful, he started to carry his own weather. He became proactive. Doing so not only lifted him but lifted everyone around him, too. Can you guess the name of the story? (Rudolph the Red-Nosed Reindeer)

Rudolph the Red-Nosed Reindeer story and song from youtube

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This is my 991st post, so I’ll tell you a little bit about the number 991:

991 is a prime number that can be written as the sum of consecutive prime numbers two different ways:
127 + 131 + 137 + 139 + 149 + 151 + 157 = 991; that’s seven consecutive primes.
191 + 193 + 197 + 199 + 211 = 991; that’s five consecutive primes.

991 is a palindrome two different ways:
33133 in BASE 4 because 3(4⁴) + 3(4³) + 1(4²) + 3(4¹) + 3(4⁰) = 991
131 in BASE 30 because 1(30²) + 3(30) + 1(1) = 991

  • 991 is a prime number.
  • Prime factorization: 991 is prime.
  • The exponent of prime number 991 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 991 has exactly 2 factors.
  • Factors of 991: 1, 991
  • Factor pairs: 991 = 1 × 991
  • 991 has no square factors that allow its square root to be simplified. √991 ≈ 31.48015

How do we know that 991 is a prime number? If 991 were not a prime number, then it would be divisible by at least one prime number less than or equal to √991 ≈ 31.5. Since 991 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 991 is a prime number.

 

983 Candy Cane

Candy canes have been a part of the Christmas season for ages. Here’s a candy cane puzzle for you to try. It’s a level 6 so it won’t be easy, but you will taste its sweetness once you complete it. Go ahead and get started!

Print the puzzles or type the solution in this excel file: 12 factors 978-985

Here’s some information about prime number 983:

983 is the sum of consecutive prime numbers two different ways:
It is the sum of the seventeen prime numbers from 23 to 97.
It is also the sum of the thirteen prime numbers from 47 to 103.

  • 983 is a prime number.
  • Prime factorization: 983 is prime.
  • The exponent of prime number 983 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 983 has exactly 2 factors.
  • Factors of 983: 1, 983
  • Factor pairs: 983 = 1 × 983
  • 983 has no square factors that allow its square root to be simplified. √983 ≈ 31.35283

How do we know that 983 is a prime number? If 983 were not a prime number, then it would be divisible by at least one prime number less than or equal to √983 ≈ 31.4. Since 983 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 983 is a prime number.

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