mode

1600 How Would You Describe This Horse Race?

/ ivasallay / Leave a comment

Today’s Puzzle:

Do most of the numbers from 1501 to 1600 have 2 factors, 4 factors, 6 factors, or what? A horse race is a fun way to find the answer to that puzzle!

As I’ve done several times before, I’ve made a horse race for this multiple of 100 and the 99 numbers before it. A horse moves when a number comes up with a particular amount of factors. Some of the races I’ve done in the past have been exciting with several lead changes. In other races, one horse ran quite quickly, leaving all other horses in the dust. One previous horse race resulted in a tie.

How will you describe this horse race? Exciting or boring? Surprizing or predictable? Pick your pony and watch the race to the end before you decide on an adjective.

Click here if you would like the Horse Race to be slightly bigger.

1501 to 1600 Horse Race

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Wow! I’ve not seen that happen before! Visually it looks like 4 won the race, but this horse race is really about finding the mode There are two modes, 4 and 8, for the amount of factors for the numbers from 1501 to 1600. It’s about which amount of factors comes up most often for the entire set of numbers, not which one of those occurs first. Thus, for that reason, I would describe the horse race above as deceptive. That horse race looked at the amount of factors five numbers at a time. Here’s what happens if we look at ten numbers at a time:

1501 to 1600 Horse Race (by tens)

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In the second horse race, it is much more clear that the race ends in a tie, and the mode is BOTH 4 and 8.

Prime Factorization of Numbers from 1501 to 1600:

Of those 100 numbers, 38 have square roots that can be simplified; 62 do not.

Factor Trees for 1600:

1600 has MANY possible factor trees. Some are symmetrical; some are not. Here are two nicely shaped ones:

Factors of 1600:

  • 1600 is a composite number and a perfect square.
  • Prime factorization: 1600 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5, which can be written 1600 = 2⁶ × 5².
  • 1600 has at least one exponent greater than 1 in its prime factorization so √1600 can be simplified. Taking the factor pair from the factor pair table below with the same number for both factors, we get
    √1600 = (√40)(√40) = 40. However, you could also use
    √1600 = (√4)(√400) = 2 × 20 = 40,
    √1600 = (√16)(√100) = 4 × 10 = 40, or
    √1600 = (√25)(√64) = 5 × 8 = 40.
  • The exponents in the prime factorization are 6 and 2. Adding one to each exponent and multiplying we get (6 + 1)(2 + 1) = 7 × 3 = 21. Therefore 1600 has exactly 21 factors.
  • The factors of 1600 are outlined with their factor pair partners in the graphic below.

More about the Number 1600:

1600 is the sum of two squares:
32² + 24² = 1600.

1600 is the hypotenuse of two Pythagorean triples:
448-1536-1600, calculated from 32² – 24², 2(32)(24), 32² + 24².
It is also (7-24-25) times 64.
960-1280-1600, which is (3-4-5) times 320.

1600 looks square in some other bases:
1600 = 1(40²) + 0(40) + 0(1), so it’s 100₄₀.
1600 =1(39²) + 2(39) + 1(1), so it’s 121₃₉.
1600 =1(38²) + 4(38) + 4(1), so it’s 144₃₈.
1600 =1(37²) + 6(37) + 9(1), so it’s 169₃₇.

Furthermore, 1600 is a repdigit in base 7:
1600 = 4(7³ + 7² + 7¹ + 7º), so it’s 4444₇.

 

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

/ ivasallay / Leave a comment

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
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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.

 

What often happens to a number like 599 when a number next to it has so many factors?

/ ivasallay / 4 Comments

The square root of any number from 576 to 624 is between 24 and 25. That means the first number in any of their factor pairs will be 24 or less. The combined number of factor pairs for the 49 integers from 576 to 624 is 189. The number of factor pairs for any given number ranges from 1 to 12. Let’s look at the averages: The mean (189/49) is 3.857 factor pairs per number. The median is 3 factor pairs, and the mode (the number of factor pairs that occurs most often) is 4 factor pairs.

Distribution of the Number of Factor Pairs from 576 to 624

What happens to a number like 599 when a number next to it has far more than the average number of factor pairs? Quite often, but not always, that number has no choice but to be a prime number.

Number of Factor Pairs around 599

Even though 600 has only three prime factors (2, 3 and 5), it still managed to be divisible by 50% of the numbers from 1 to 24, and there just aren’t many possibilities left for the numbers immediately before or after it.

The twelve numbers less than or equal to 24 that will divide into 600 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20 and 24.

Because 9, 14, 16, 18, 21, and 22 each have 2 or 3 as a prime factor, those six numbers also are not possible factors of 599 or 601.

Every whole number is divisible by 1, but besides that, there are only six numbers available as possible factors for those two numbers: 7, 11, 13, 17, 19, and 23. Since neither 599 nor 601 is divisible by any of those numbers, they turn out to be twin primes.

Usually at least one of the numbers before or after a number with far more than its fair share of factor pairs will be a prime number.

119 and 121, the numbers before and after 120 are notable exceptions. √120 ≈ 10.95441. The factors of 120 that are less than or equal to 10 are 1, 2, 3, 4, 5, 6, 8, and 10 which is 80% of the possible factors. Yet 119 managed to be divisible by 7, and 121 managed to be divisible by 11 so neither one of them is a prime number.

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If you add up the digits of 499, 589, or 598, you will get 22.

599 is the smallest whole number whose digits add up to 23. Thank you, OEIS.org for that number fact.

  • 599 is a prime number and a twin prime with 601.
  • Prime factorization: 599 is prime.
  • The exponent of prime number 599 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 599 has exactly 2 factors.
  • Factors of 599: 1, 599
  • Factor pairs: 599 = 1 x 599
  • 599 has no square factors that allow its square root to be simplified. √599 ≈ 24.4744765

How do we know that 599 is a prime number? If 599 were not a prime number, then it would be divisible by at least one prime number less than or equal to √599 ≈ 24.5. Since 599 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, or 23, we know that 599 is a prime number.