Author: ivasallay

917 How to Solve for X with Candy

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How many m&m’s are there in one fun size Halloween pack of m&m’s?

I don’t know. You don’t know. Nobody knows. That’s why these little fun size packs make the perfect UNKNOWN. For this activity, there may even be a negative number of m&m’s in a pack because I’m only using blue and orange m&m’s, and I’m letting each blue m&m equal negative one and each orange m&m equal positive one.

I’m also letting the front side of the fun size package equal negative x, and the back side equal positive x. In algebra we often call our unknown x.

(The colors chosen don’t matter, as long as there are only two of them, and you are consistent with that color being positive or negative. The front of the package could just as easily be +1 and the back -1. Consistency is important. Choose the values you want to use and stick with them. You can also use ALL the m&m’s in a few single packs and have the side of the candy with the m be positive and the side without the m be negative. You can use the empty wrappers as x and -x.)

We can figure out how many m&m’s are in the pack by balancing an equation. The number of m&m’s in a pack is x. We will solve for x by using the very best algebra tiles in the world, m&m’s!

Besides fun size m&m’s (or Skittles or Reece’s Pieces) we need a paper balance for our equations:

Click Equation Balance for a printable pdf of the paper balance.

Now let’s solve x – 3 = 5 by using the m&m’s to find x. This is how the equation balance should look to begin:

We want to get the wrapper by itself, so what do we do? To keep the equation balanced, we add three (positive) orange m&m’s (one at a time) to both sides of the paper balance:

Three (negative) blue m&m’s plus three (positive) orange m&m’s are equal to zero, so we can remove them.

Mmm. I just ate zero m&m’s, and they tasted so good! That leaves us with x = 8, so we have found x, and the equation has been solved! (If you have more equations to solve, you might want to wait to eat the m&m’s until you’re just about finished.)

Now let’s try finding x when the equation is a little more complicated, -2x = x + 12. This is how the balance should look at the beginning:

We want to get all the x’s on one side so we subtract x from both sides of the equation by adding a (negative) front-facing wrapper to both sides of the balance.

Since x – x = 0, we can remove the front-facing and back-facing wrappers from the right side of the equation:

We can arrange the 12 candies into 3 rows of 4.

Now we can divide both sides of the equation by 3.

All that’s left to do is change the signs of EVERYTHING on both sides of the equation:

Thus x = -4. We solved for x correctly because we kept the equation balanced every step of the way.

Now let me tell you a little bit about the number 917:

917 is the sum of five consecutive prime numbers:
173 + 179 + 181 + 191 + 193 = 917

Rearrange its digits and 917 becomes 197 in BASE 26.

  • 917 is a composite number.
  • Prime factorization: 917 = 7 × 131
  • 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 917 has exactly 4 factors.
  • Factors of 917: 1, 7, 131, 917
  • Factor pairs: 917 = 1 × 917 or 7 × 131
  • 917 has no square factors that allow its square root to be simplified. √917 ≈ 30.282007859

916 Witch’s Hat

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Today’s puzzle looks a little like a witch’s hat. Solving it could be bewitching. It contains a lot of perfect squares, and it’s only a level 2. Give it a try! Just make sure you don’t write a number greater than ten in the top row or the first column. Use all the numbers from one to ten in both places. There is only one solution.

Print the puzzles or type the solution on this excel file: 10-factors-914-922

916 looks the same upside-down as it does right-side up, so it is a Strobogrammatic number.

It looks like someone put a spell on that number!

Not only that, in BASE 26, our 916 looks like 196. There is a whole lot of magic going on here!

30² + 4² = 916, so 916 is the hypotenuse of a Pythagorean triple:
240-884-916, which can be calculated from 2(30)(4), 30² – 4², 30² + 4².

  • 916 is a composite number.
  • Prime factorization: 916 = 2 × 2 × 229, which can be written 916 = 2² × 229
  • 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 916 has exactly 6 factors.
  • Factors of 916: 1, 2, 4, 229, 458, 916
  • Factor pairs: 916 = 1 × 916, 2 × 458, or 4 × 229
  • Taking the factor pair with the largest square number factor, we get √916 = (√4)(√229) = 2√229 ≈ 30.2654919

 

 

915 Traditional Vampire Deterrent

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915 is the hypotenuse of four Pythagorean triples:

  • 165-900-915 which is 15 times (11-60-61)
  • 408-819-915 which is 3 times (136-273-305)
  • 621-672-915 which is 3 times (207-224-305)
  • 549-732-915 which is (3-4-5) times 183

 

Print the puzzles or type the solution on this excel file: 10-factors-914-922

Here’s a little more about the number 915:

915 is repdigit 555 in BASE 13 because 5(13²) + 5(13) + 5(1) = 5(183) = 915.

It is palindrome 393 in BASE 16 because 3(16²) + 9(16) +3(1) = 915.

And as 195 in BASE 26, it uses its base 10 digits in a different order. Note that 1(26²) + 9(26) + 5(1) = 915.

  • 915 is a composite number.
  • Prime factorization: 915 = 3 × 5 × 61
  • 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 915 has exactly 8 factors.
  • Factors of 915: 1, 3, 5, 15, 61, 183, 305, 915
  • Factor pairs: 915 = 1 × 915, 3 × 305, 5 × 183, or 15 × 61
  • 915 has no square factors that allow its square root to be simplified. √915 ≈ 30.2489669.

 

914 Jack-O’Lantern Mystery Level

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25² + 17² = 914, so 914 is the hypotenuse of a Pythagorean triple:
336-850-914, which is 2 times (168-425-457).
You can also calculate it from 2(25)(17), 25² – 17², 25² + 17².

914 becomes 194 in BASE 26. Notice that both bases use the same digits.

914 is also palindrome 494 in BASE 14.

Today’s puzzle is a jack o’lantern that will kick off two weeks worth of Halloween themed puzzles.

Print the puzzles or type the solution on this excel file: 10-factors-914-922

  • 914 is a composite number.
  • Prime factorization: 914 = 2 × 457
  • 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 914 has exactly 4 factors.
  • Factors of 914: 1, 2, 457, 914
  • Factor pairs: 914 = 1 × 914 or 2 × 457
  • 914 has no square factors that allow its square root to be simplified. √914 ≈ 30.2324

913 Haunted Ten Frame House for Ten Timid Ghosts

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Last year right before Halloween, I read a wonderful book to a class of kindergartners. The book was Ten Timid Ghosts by Jennifer O’Connell. The students drew haunted houses and a few trees. As I read the story, they used ten bean counters to show where each ghost was during the story. At any given time each ghost was either in the house or in the woods. They could see lots of number sentences as the story progressed.

This year I liked the idea of making a haunted house with ten window frames and ten sections for the trees in the woods. That way each of the ten ghosts could have a specific place to be throughout the story. I made the ten frames vertical to make odd and even amounts easier to see. There seems to be some ghostly figures flying around the tree branches, too.

You are welcome to use this haunted house as you add and subtract ghosts from the house and the woods. You’ll need some ghosts, too. Dry great northern beans are white and make great ghosts. If you like, you can add eyes and a mouth to the ghostly beans with a marker.

For those who would rather use less printer ink, I’ve made a similar picture that can be colored after it is printed:

You may also want to look at another activity that uses this same book. Sara Gast’s power point was especially made for kids with autism, but other children would enjoy it as well. You can use her activity to reinforce the concept of ordinal numbers, too.

Young children everywhere are sure to enjoy this fun book as they learn to count and to add and subtract.

Now I’ll write a little bit about the number 913.

If you’ve read my last few posts, you may have noticed that 910, 911, and 912 can each be represented in BASE 26 using their base ten digits in a different order. That pattern continues for 913. In fact, it is true for 914 – 919, as well. However, 913 is even more special because not only is it 193 in BASE 26, it is also 391 in BASE 16. Thank you, OEIS.org for that fun fact.

9 – 1 + 3 = 11, so 913 is divisible by eleven.

  • 913 is a composite number.
  • Prime factorization: 913 = 11 × 83
  • 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 913 has exactly 4 factors.
  • Factors of 913: 1, 11, 83, 913
  • Factor pairs: 913 = 1 × 913 or 11 × 83
  • 913 has no square factors that allow its square root to be simplified. √913 ≈ 30.2158899

 

912 and Level 6

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912 is the sum of the ten prime numbers from 71 to 109.

It is also the sum of these four consecutive primes:

  • 223 + 227 + 229 + 233 = 912

912 is 192 in BASE 26 because 1(26²) + 9(26) + 2(1) = 912.

Print the puzzles or type the solution on this excel file: 12 factors 905-913

  • 912 is a composite number.
  • Prime factorization: 912 = 2 × 2 × 2 × 2 × 3 × 19, which can be written 912 = 2⁴ × 3 × 19
  • The exponents in the prime factorization are 4, 1 and 1. Adding one to each and multiplying we get (4 + 1)(1 + 1)(1 + 1) = 5 × 2 × 2 = 20. Therefore 912 has exactly 20 factors.
  • Factors of 912: 1, 2, 3, 4, 6, 8, 12, 16, 19, 24, 38, 48, 57, 76, 114, 152, 228, 304, 456, 912
  • Factor pairs: 912 = 1 × 912, 2 × 456, 3 × 304, 4 × 228, 6 × 152, 8 × 114, 12 × 76, 16 × 57, 19 × 48 or 24 × 38
  • Taking the factor pair with the largest square number factor, we get √912 = (√16)(√57) = 4√57 ≈ 30.1993377.

911 Mystery Level Puzzle

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911 is a prime number that is also the sum of three consecutive primes:

  • 293 + 307 + 311 = 911

911 is 191 in BASE 26 because 1(26²) + 9(26) + 1(1) = 911

911 is palindrome 12121 in BASE 5 because 1(5⁴) + 2(5³) + 1(5²) + 2(5) + 1(1) = 911

Print the puzzles or type the solution on this excel file: 12 factors 905-913

  • 911 is a prime number.
  • Prime factorization: 911 is prime.
  • The exponent of prime number 911 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 911 has exactly 2 factors.
  • Factors of 911: 1, 911
  • Factor pairs: 911 = 1 × 911
  • 911 has no square factors that allow its square root to be simplified. √911 ≈ 30.1827765

How do we know that 911 is a prime number? If 911 were not a prime number, then it would be divisible by at least one prime number less than or equal to √911 ≈ 30.2. Since 911 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 911 is a prime number.

 

 

 

 

910 and Level 5

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910 is the hypotenuse of four Pythagorean triples:

  • 224-882-910, which is 14 times (16-63-65).
  • 350-840-910, which is (5-12-13) times 70.
  • 462-784-910, which is 14 times (33-56-65)
  • 546-728-910, which is (3-4-5) times 182.

Print the puzzles or type the solution on this excel file: 12 factors 905-913

910 is 190 in BASE 26, and 910 looks interesting in some other bases, too:

  • 4114 in BASE 6, because 4(6³) + 1(6²) + 1(6¹) + 4(6º) = 910
  • 1221 in BASE 9, because 1(9³) + 2(9²) + 2(9¹) + 1(9º) = 910
  • QQ in BASE 34 (Q is 26 base 10), because 26(34¹) + 26(34º) = 26(34 + 1) = 26(35) = 910
  • 26 0 BASE 35, because 26(35) + 0(1) = 26(35) = 910

What are the factors of 910?

  • 910 is a composite number.
  • Prime factorization: 910 = 2 × 5 × 7 × 13
  • 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 910 has exactly 16 factors.
  • Factors of 910: 1, 2, 5, 7, 10, 13, 14, 26, 35, 65, 70, 91, 130, 182, 455, 910
  • Factor pairs: 910 = 1 × 910, 2 × 455, 5 × 182, 7 × 130, 10 × 91, 13 × 70, 14 × 65, or 26 × 35
  • 910 has no square factors that allow its square root to be simplified. √910 ≈ 30.166206

 

909 and Level 4

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909 is a palindrome in base 10 because 9(100) + 0(10) + 9(1) = 909.

It is also palindrome 757 in BASE 11 because 7(121) + 5(11) + 7(1) = 909.

Print the puzzles or type the solution on this excel file: 12 factors 905-913

Here’s a little more about the number 909:

30² + 3² = 909

909 is the hypotenuse of Pythagorean triple 180-891-909, which is 9 times (20-99-101) and can be calculated from 2(30)(3), 30² – 3², 30² + 3²

As you know, 909 is made with 9’s and 0’s. OEIS.org informs us that 909 times 2, 3, 4, 5, 6, 7, 8, or 9 do NOT contain even one 9 or 0. That’s a little spooky, but see for yourself:

  • 909 is a composite number.
  • Prime factorization: 909 = 3 × 3 × 101, which can be written 909 = 3² × 101
  • 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 909 has exactly 6 factors.
  • Factors of 909: 1, 3, 9, 101, 303, 909
  • Factor pairs: 909 = 1 × 909, 3 × 303, or 9 × 101
  • Taking the factor pair with the largest square number factor, we get √909 = (√9)(√101) = 3√101 ≈ 30.14962686

908 Haunted House

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I combined most of this week’s puzzles to make a haunted house. None of the levels are listed on it, but I will include the levels as I publish each puzzle separately sometime this week. If you enter this haunted house, will you be able to escape? Warning: This should not be the first Find The Factors puzzle you try. A couple of the puzzles in it may be easy, but the rest could be very scary!

Print the puzzles or type the solution on this excel file: 12 factors 905-913

908 can be written as the sum of two consecutive odd numbers:

  • 453 + 455 = 908

Four consecutive even numbers:

  • 224 + 226 + 228 + 230 = 908

And eight consecutive counting numbers:

  • 110 + 111 + 112 + 113 + 114 + 115 + 116 + 117 = 908

There wasn’t much to say about the number 908, but it’s always possible to find something.

  • 908 is a composite number.
  • Prime factorization: 908 = 2 × 2 × 227, which can be written 908 = 2² × 227
  • 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 908 has exactly 6 factors.
  • Factors of 908: 1, 2, 4, 227, 454, 908
  • Factor pairs: 908 = 1 × 908, 2 × 454, or 4 × 227
  • Taking the factor pair with the largest square number factor, we get √908 = (√4)(√227) = 2√227 ≈ 30.1330383