how many numbers between 100-200 are not divisible by 4 and 5
Answers
Answer:
We need to find the sum of numbers between 100 and 200 which are not divisible
We can first find the sum of all numbers between 100 and 200, then later subtract the sum of numbers divisible
Let S be sum of numbers between 100 and 200.
S=
2
200×201
−
2
100×101
{Using the formula sum of series 1,2,3,...,n is
2
n(n+1)
}
⇒S=15050
Let s be sum of numbers between 100 and 200 which are divisible
s=
2
20
(2×105+(20−1)×5)=3050
The required sum =S−s=15050−3050=12000
Since 100 and 200 are both divisible by 5, I will solve this as being inclusive, because the inclusive answer will be the same as the exclusive one.
The number of integers between 2 integer values inclusive is the difference of the 2 integers plus 1. Therefore, there are 200−100+1=101200−100+1=101 integers between 100 and 200 inclusive.
To determine the count that are divisible by a number, you take the upper limit divided by the number and round down. Then you take the lower limit divided by the number and round up. Then you count how many integers are in that interval.
For example, the number of numbers between 100 and 200 divisible by 3 would be determined as thus:
200/3≈66.7200/3≈66.7 , which rounds down to 66.
100/3≈33.3100/3≈33.3 , which rounds up to 34.
Therefore, there are 66−34+1=3366−34+1=33 numbers between 100 and 200 divisible by 3.
Using that same process, there are 26 numbers between 100 and 200 divisible by 4, and 21 numbers between 100 and 200 divisible by 5.
It is possible for some numbers to be divisible by more than 1 of those, so we also need to know those possibilities.
For a number to be divisible by both 3 and 4, it must also be divisible by their LCM, 12. Using the same process as before, there are 8 numbers between 100 and 200 divisible by 12.
For a number to be divisible by both 3 and 5, it must be divisible by 15. Therefore, there are 7 numbers between 100 and 200 divisible by 15.
For 4 and 5, there are 6 numbers divisible by both.