Find how many integers between 200 and 500 are divisible by 8
You can solve this problem as an Arithmetic Progression.
The series: 208, 216, 224,...., 488, 496
a(n) = 496
a(1) = 208
d = 8
For an A.P.,
a(n) = a(1) + (n-1)d
=> 496 = 208 + (n-1)8
=> 288/8 = n - 1
=> n = 36 + 1 = 37
Hence, there are 37 such integers.
Find the first number in thist interval that is divisible by 8
In this case:
200 / 8 = 25
Find the last number in thist interval that is divisible by 8
In this case:
496 / 8 = 62
Numbers 200 and 496 assume arithmetic progression, where first member a1 = 200 and last member a62 = 496
Common difference is d = 8
In arithmetic progression:
an = a1 + ( n - 1 ) d
In this case:
a62 = a1 + ( n - 1 ) d
496 = 200 + ( n - 1 ) ∙ 8
496 = 200 + 8 n - 8
496 = 192 + 8 n
Subtract 192 to both sides
496 - 192 = 192 + 8 n - 192
304 = 8 n
Divide both sides by 8
38 = n
n = 38
200 shouldn't be included, should it?
The question says 'between 200 and 500', so.
To find the number of integers between 200 and 500 that are divisible by 8, we can use the following steps:
1. Find the smallest integer greater than or equal to 200 that is divisible by 8. We can do this by dividing 200 by 8 and rounding up to the nearest whole number: ceil(200/8) = 25. ceil() is a function that rounds up to the nearest integer.
2. Find the largest integer less than or equal to 500 that is divisible by 8. We can do this by dividing 500 by 8 and rounding down to the nearest whole number: floor(500/8) = 62. floor() is a function that rounds down to the nearest integer.
3. Subtract the result from step 1 from the result of step 2 and add 1 to get the number of integers between 200 and 500 that are divisible by 8: 62 - 25 + 1 = 38.
Therefore, there are 38 integers between 200 and 500 that are divisible by 8.