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

903 Mystery Level

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903 can be written as the sum of consecutive numbers seven different ways:

  • 451 + 452 = 903; that’s 2 consecutive numbers.
  • 300 + 301 + 302 = 903; that’s 3 consecutive numbers.
  • 147 + 148 + 149 + 150 + 151 + 153 = 903; that’s 6 consecutive numbers.
  • 126 + 127 + 128 + 129 + 130 + 131 + 132 = 903; that’s 7 consecutive numbers.
  • 58 + 59 + 60 + 61 + 62 + 63 + 64 + 65 + 66 + 67 + 68 + 69 + 70 + 71 = 903; that’s fourteen consecutive numbers.
  • 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 + 43 + 44 + 45 + 46 + 47 + 48 + 49 + 50 + 51 + 52 + 53 = 903; that’s 21 consecutive numbers.
  • 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31 + 32 + 33 + 34 + 35 + 36 + 37 + 38 + 39 + 40 + 41 + 42 = 903; that’s 42 consecutive numbers.

That last way means that 903 is the 42nd triangular number. It happened because (42 × 43)/2 = 903.

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

  • 903 is a composite number.
  • Prime factorization: 903 = 3 × 7 × 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 903 has exactly 8 factors.
  • Factors of 903: 1, 3, 7, 21, 43, 129, 301, 903
  • Factor pairs: 903 = 1 × 903, 3 × 301, 7 × 129, or 21 × 43
  • 903 has no square factors that allow its square root to be simplified. √903 ≈ 30.0499584.

902 and Level 5

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902 is the hypotenuse of a Pythagorean triple:

198-880-902 which is 22 times (9-40-41)

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

  • 902 is a composite number.
  • Prime factorization: 902 = 2 × 11 × 41
  • 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 902 has exactly 8 factors.
  • Factors of 902: 1, 2, 11, 22, 41, 82, 451, 902
  • Factor pairs: 902 = 1 × 902, 2 × 451, 11 × 82, or 22 × 41
  • 902 has no square factors that allow its square root to be simplified. √902 ≈ 30.0333.

901 and Level 4

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

901 is the 25th centered triangular number because (23×24 + 24×25 + 25×26)/2 = 901. That is the same as saying that 901 is the sum of the 23rd, the 24th, and the 25th triangular numbers.

901 is the sum of two squares two different ways:

  • 30² + 1² = 901
  • 26² + 15² = 901

901 is the hypotenuse of FOUR Pythagorean triples:

  • 60-899-901, calculated from 2(30)(1), 30² – 1², 30² + 1²
  • 424-795-901, which is (8-15-17) times 53
  • 451-780-901, calculated from 26² – 15², 2(26)(15), 26² + 15²
  • 476-765-901, which is 17 times (28-45-53)

Two of those were primitives. That can only happen because ALL of 901’s prime factors are Pythagorean triple hypotenuses.

  • 901 is a composite number.
  • Prime factorization: 901 = 17 × 53
  • 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 901 has exactly 4 factors.
  • Factors of 901: 1, 17, 53, 901
  • Factor pairs: 901 = 1 × 901 or 17 × 53
  • 901 has no square factors that allow its square root to be simplified. √901 ≈ 30.016662

900 Pick Your Pony. Who’ll Win This Amount of Factors Horse Race?

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I really like this rhyme that I saw for the first time this week (even though it’s all over the net):

Hey diddle diddle, the median’s the middle,
You add then divide for the mean.
The mode is the one that appears the most,
And the range is the difference between.

All of the numbers from 801 to 900 have at least 2 factors, but no more than 32 factors. 32 – 2 = 30, so 30 is the range of the amount of factors.

There are 100 numbers from 801 to 900. If you list the amount of factors for each number, then arrange those amounts from smallest to largest, the amounts that will appear in the 50th and 51st spots will both be 6. That means that 6 is the median amount of factors. If we had different amounts in the 50th and 51st spots, we would average the two amounts together to get the median.

If you add up the amounts of factors that the numbers from 801 to 900 have, you will get 794. If you divide 794 by 100, the number of entries, then you will know that 7.94 is the mean amount of factors.

What about the mode? Which amount of factors appears the most? That’s why we are having a Horse Race, to see if more numbers have 2 factors, 3 factors, 4 factors, or a different amount of factors. So pick your pony. We’ll see which amount wins, and we’ll find out what the mode is at the same time.

The contenders are these amounts: 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 27, 32.

I should tell you that only perfect squares can have an odd amount of factors, so you probably don’t want to pick an odd amount.

Here are some interesting facts about the numbers from 801 to 900 that might help you decide which pony to pick.

  • We had the smallest two consecutive numbers with exactly 12 factors: (819, 820)
  • We had the fourth prime decade: (821, 823, 827, 829). All four of those numbers are prime numbers and have exactly two factors.
  • We had five consecutive numbers whose square roots can be reduced: (844, 845, 846, 847, 848). Three of those numbers had 6 factors, one had 10, and one had 12.
  • We also had 840, the smallest number with exactly 32 factors
  • 900 is the smallest number with exactly 27 factors. Coincidentally, the amount that is the mode will appear 27 times.

As the following table shows, there are 42 integers from 801 to 900 that have square roots that can be simplified. 42 is more than any previous set of 100 numbers has given us. Even still we are still holding close to just under 40% of integers having square roots that can be simplified.

Okay. If you’ve picked your pony, NOW you can watch the Horse Race:

900 Horse Race
make science GIFs like this at MakeaGif
Hmm…

The race was exciting for a second or two.

As you can see from the Horse Race the mode is 4. How did your pony do?

Here’s a little more about the number 900:

900 is the sum of the fourteen prime numbers from 37 to 97.

24² + 18² = 900

900 is the hypotenuse of two Pythagorean triples:

  • 252-864-900, which is 24² – 18², 2(24)(18), 24² + 18². It is also (7-24-25) times 36.
  • 540-720-900, which is (3-4-5) times 180.

900 is the sum of the interior angles of a heptagon (seven-sided polygon).

  • 900 is a composite number and a perfect square.
  • Prime factorization: 900 = 2 × 2 × 3 × 3 × 5 × 5, which can be written 900 = 2² × 3² × 5²
  • The exponents in the prime factorization are 2, 2 and 2. Adding one to each and multiplying we get (2 + 1)(2 + 1)(1 + 1) = 3 × 3 × 3 = 27. Therefore 900 has exactly 27 factors.
  • Factors of 900: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 150, 180, 225, 300, 450, 900
  • Factor pairs: 900 = 1 × 900, 2 × 450, 3 × 300, 4 × 225, 5 × 180, 6 × 150, 9 × 100, 10 × 90, 12 × 75, 15 × 60, 18 × 50, 20 × 45, 25 × 36, or 30 × 30
  • Taking the factor pair with the largest square number factor, we get √900 = (√30)(√30) = 30.

 

899 and Level 3

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Yesterday I worked with a student who knew that 5 × 5 = 25 but couldn’t remember what 4 × 6 is. I didn’t use the variable “n”, but I used some examples from the times table to help her understand that (n-1)(n+1) = n² – 1. Always. Then I said to her, “30 × 30 = 900, so how much is 29 × 31?” She answered, “899?” I told her, “Yes, it is.” She was pretty pleased with herself.

29 and 31 are twin primes, so that makes 29 × 31 = 899 even cooler.

899 is also the hypotenuse of a Pythagorean triple:

  • 620-651-899 which is (20-21-29) times 31.

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

  • 899 is a composite number.
  • Prime factorization: 899 = 29 × 31
  • 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 899 has exactly 4 factors.
  • Factors of 899: 1, 29, 31, 899
  • Factor pairs: 899 = 1 × 899 or 29 × 31
  • 899 has no square factors that allow its square root to be simplified. √899 ≈ 29.9833

898 and Level 2

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27² + 13² = 898. That means that 898 is the hypotenuse of a Pythagorean triple:

  • 560-702-898, which is 2 times (280-351-449).

898 reads the same way frontwards and backwards so it is a palindrome in base 10.

It is also palindrome 747 in BASE 11 because 7(11²) + 4(11) + 7(1) = 898

AND it is palindrome 1G1 in BASE 23 (G is 16 in base 10) because 1(23²) + 16(23) +1(1) =898.

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

  • 898 is a composite number.
  • Prime factorization: 898 = 2 × 449
  • 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 898 has exactly 4 factors.
  • Factors of 898: 1, 2, 449, 898
  • Factor pairs: 898 = 1 × 898 or 2 × 449
  • 898 has no square factors that allow its square root to be simplified. √898 ≈ 29.9666