Question
How many divisor of 6 numbers are there between 100 and 1000?
Answers
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We need to count the number of divisors of all the numbers between 100 and 1000, inclusive. We can start by listing out all the divisors of the smallest number, which is 100:
1, 2, 4, 5, 10, 20, 25, 50, 100.
There are 9 divisors of 100. Notice that if we multiply 100 by any prime number, we will get a number between 100 and 1000 with at least 10 divisors. For example, if we multiply 100 by 7, we get 700, which has the following divisors:
1, 2, 4, 5, 7, 10, 14, 20, 25, 35, 50, 70, 100, 175, 350, 700.
There are 16 divisors of 700. We can repeat this process for all the prime numbers between 2 and 31 (the largest prime number less than or equal to the square root of 1000). This gives us a total of 10 numbers with 9 divisors each, and 24 numbers with 16 divisors each.
Finally, we need to consider the numbers that have more than 16 divisors. These numbers must have at least 3 distinct prime factors. We can find all such numbers by listing out all the products of 3 distinct primes less than or equal to 31 (since any larger prime would result in a number greater than 1000). There are a total of 53 such products, but we need to eliminate the duplicates. For example, 2 x 3 x 5 is the same as 3 x 5 x 2 and 5 x 2 x 3. After eliminating the duplicates, we are left with 36 additional numbers with more than 16 divisors each.
Therefore, the total number of divisors of all the numbers between 100 and 1000 is:
10 x 9 + 24 x 16 + 36 x (more than 16)
We don't need to know the exact number of divisors of these additional 36 numbers, only that they each have more than 16. Therefore, we can conservatively estimate that they each have 17 divisors. This gives us:
90 + 384 + 612 = 1086
Therefore, there are a total of 1086 divisors of the numbers between 100 and 1000.
1, 2, 4, 5, 10, 20, 25, 50, 100.
There are 9 divisors of 100. Notice that if we multiply 100 by any prime number, we will get a number between 100 and 1000 with at least 10 divisors. For example, if we multiply 100 by 7, we get 700, which has the following divisors:
1, 2, 4, 5, 7, 10, 14, 20, 25, 35, 50, 70, 100, 175, 350, 700.
There are 16 divisors of 700. We can repeat this process for all the prime numbers between 2 and 31 (the largest prime number less than or equal to the square root of 1000). This gives us a total of 10 numbers with 9 divisors each, and 24 numbers with 16 divisors each.
Finally, we need to consider the numbers that have more than 16 divisors. These numbers must have at least 3 distinct prime factors. We can find all such numbers by listing out all the products of 3 distinct primes less than or equal to 31 (since any larger prime would result in a number greater than 1000). There are a total of 53 such products, but we need to eliminate the duplicates. For example, 2 x 3 x 5 is the same as 3 x 5 x 2 and 5 x 2 x 3. After eliminating the duplicates, we are left with 36 additional numbers with more than 16 divisors each.
Therefore, the total number of divisors of all the numbers between 100 and 1000 is:
10 x 9 + 24 x 16 + 36 x (more than 16)
We don't need to know the exact number of divisors of these additional 36 numbers, only that they each have more than 16. Therefore, we can conservatively estimate that they each have 17 divisors. This gives us:
90 + 384 + 612 = 1086
Therefore, there are a total of 1086 divisors of the numbers between 100 and 1000.