How many divisors of 6 are found between 100 and 1000?
The divisors of 6 are 1, 2, 3, and 6.
To find how many of these divisors are between 100 and 1000, we need to find the multiples of each divisor within that range.
For 1, every number between 100 and 1000 is a multiple, so we don't need to count those.
For 2, the first multiple within the range is 100 + (2 - 100%2) = 100 + 0 = 100. The last multiple is 1000 - 1000%2 = 1000 - 0 = 1000. Since there are 451 even numbers within this range (counting both endpoints), there are 451/2 = 225 multiples of 2.
For 3, the first multiple within the range is 102 (since 100 is not a multiple of 3), and the last multiple is 999. To find how many multiples there are, we can divide the difference between the endpoints by 3 and round up: (999 - 102)/3 + 1 = 299.
For 6, the first multiple within the range is 102 (since 100 is not a multiple of 6), and the last multiple is 996. To find how many multiples there are, we can divide the difference between the endpoints by 6 and round up: (996 - 102)/6 + 1 = 150.
Adding up the number of multiples for each divisor, we get: 1 + 225 + 299 + 150 = 675.
Therefore, there are 675 divisors of 6 between 100 and 1000.
To find the number of divisors of 6 between 100 and 1000, we need to count the numbers within this range that are divisible by 6.
Step 1: Find the smallest multiple of 6 within the range:
The smallest multiple of 6 within the range is 102 (since 100 is not divisible by 6).
Step 2: Find the largest multiple of 6 within the range:
The largest multiple of 6 within the range is 996 (since 1000 is not divisible by 6).
Step 3: Calculate the number of multiples of 6 within the range:
To find the number of multiples, we take the difference between the largest and smallest multiple, and divide by 6, then add 1 to account for the inclusive range.
Number of multiples = (largest multiple - smallest multiple) / 6 + 1
= (996 - 102) / 6 + 1
= 894 / 6 + 1
≈ 149 + 1
= 150
Therefore, there are 150 divisors of 6 between 100 and 1000.