factor pairs

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.

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

907 and Level 3

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88 is 8 × 11, and 24 is 2 × 12, 3 × 8, or 4 × 6. Those are all their factor pairs in which both factors are less than or equal to 12. There is only one number that is a common factor for both 88 and 24. Write that number in the top cell of the first column and the rest of 88’s factor pair directly above 88 and the rest of 24’s factor pair directly above 24 in the top row.

Next think of a factor pair for 80 in which both factors are less than or equal to 12. You probably thought of 8 × 10, the only factor pair that qualifies. The first column already has an 8, so this 8 must go in the top row above 80. Write 10 in the first column.

The next row doesn’t have a clue, but you already have enough information to write what number must go in the first column. (Hint: it is a number that is already in the top row and can’t go in any other cell in the first column.) If you cannot figure out what goes in this cell, skip that row until later, and figure out what goes in the next cells continuing from the top cell of the first column to the bottom cell. You will fill out the top row at the same time, but each factor 1- 12 will be written above its appropriate clue instead of in order from left to right. Good luck!

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

907 is the first prime number since 887. We will not have to wait nearly as long for the next prime number. It will be 911.

907 is palindrome 32023 in BASE 4 because 3(4⁴) + 2(4³) + 0(4²) + 2(4¹) + 3(4º) = 907.

  • 907 is a prime number.
  • Prime factorization: 907 is prime.
  • The exponent of prime number 907 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 907 has exactly 2 factors.
  • Factors of 907: 1, 907
  • Factor pairs: 907 = 1 × 907
  • 907 has no square factors that allow its square root to be simplified. √907 ≈ 30.11644

How do we know that 907 is a prime number? If 907 were not a prime number, then it would be divisible by at least one prime number less than or equal to √907 ≈ 30.1. Since 907 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, or 29, we know that 907 is a prime number.

906 and Level 2

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906 looks the same upside-down so it is a strobogrammatic number.

We can tell just by looking at 906 that it is divisible by 1, 2, and 3. That means it can also be evenly divided by 6.

906 is the sum of consecutive prime numbers: 449 + 457 = 906

It is also palindrome 636 in BASE 12 because 6(144) + 3(12) + 6(1) = 906.

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

  • 906 is a composite number.
  • Prime factorization: 906 = 2 × 3 × 151
  • 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 906 has exactly 8 factors.
  • Factors of 906: 1, 2, 3, 6, 151, 302, 453, 906
  • Factor pairs: 906 = 1 × 906, 2 × 453, 3 × 302, or 6 × 151
  • 906 has no square factors that allow its square root to be simplified. √906 ≈ 30.099833887

 

905 and Level 1

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905 is the sum of the seventeen prime numbers from 19 to 89.

905 is also the sum of these seven consecutive prime numbers:

  • 109 + 113 + 127 + 131 + 137 + 139 + 149 = 905

Here’s today’s puzzle:

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

29² + 8² = 905, and 28² + 11² = 905, making 905 the hypotenuse of four Pythagorean triples:

  • 95-900-905, which is 5 times (19-180-181)
  • 464-777-905, computed from 2(29)(8), 29² – 8², 29² + 8²
  • 543-724-905, which is (3-4-5) times 181
  • 616-663-905, computed from 2(28)(11), 28² – 11², 28² + 11²

The numbers in red are factors of 905.

  • 905 is a composite number.
  • Prime factorization: 905 = 5 × 181
  • 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 905 has exactly 4 factors.
  • Factors of 905: 1, 5, 181, 905
  • Factor pairs: 905 = 1 × 905 or 5 × 181
  • 905 has no square factors that allow its square root to be simplified. √905 ≈ 30.0832179

904 and Level 6

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30² + 2² = 904

That means 904 is the hypotenuse of a Pythagorean triple:

  • 120-896-904 which is 8 times (15-112-113)

Print the puzzles or type the solution on this excel file: 10-factors-897-904

  • 904 is a composite number.
  • Prime factorization: 904 = 2 × 2 × 2 × 113, which can be written 904 = 2³ × 113
  • The exponents in the prime factorization are 3 and 1. Adding one to each and multiplying we get (3 + 1)(1 + 1) = 4 × 2 = 8. Therefore 904 has exactly 8 factors.
  • Factors of 904: 1, 2, 4, 8, 113, 226, 452, 904
  • Factor pairs: 904 = 1 × 904, 2 × 452, 4 × 226, or 8 × 113
  • Taking the factor pair with the largest square number factor, we get √904 = (√4)(√226) = 2√226 ≈ 30.06659