find how many integers between 100 and 400 are divisible by 8
Answers
Step-by-step explanation:
Given :-
The numbers are 100 and 400
To find :-
Find how many integers between 100 and 400 are divisible by 8 ?
Solution :-
Method -1:-
Given numbers are 100 and 400
The list of integers between 100 and 400
= 101,102,...,399.
The list of integers between 100 and 400 which are divisible by 8
= 104, 112, 120, ..., 392.
First term (a) = 104
Common difference = d
=112-104 = 8
= 120-112 = 8
Since the common difference is same throughout the series
They are in the Arithmetic Progression.
The last term = 392
Let an = 392
We know that
The nth term of an AP =an = a+(n-1)d
We have,
a = 104
d = 8
an = 392
On Substituting these values in the above formula then
=> 104+(n-1)(8) = 392
=> 104+8n-8 = 392
=> 8n+96 = 392
=> 8n = 392-96
=> 8n = 296
=> n = 296/8
=> n = 37
Number of terms = 37
Method -2:-
Given numbers are 100 and 400
Let a = 100
Let b = 400
The integer which is divisible by 8 then the common difference between every two consecutive integers = 8
d = 8
Let the number of AM's between two numbers be n
We know that
d = (b-a)/(n+1)
=> 8 =(400-100)/(n+1)
=> 8 = 300/(n+1)
=> 8(n+1) = 300
=> 8n+8 = 300
=> 8n = 300-8
=> 8n = 292
=> n = 292/8
=> n = 36.5 ~ 37
=> n = 37
Since n must be a natural number.
So the required integers = 37
Answer :-
The number of integers between 100 and 400 which are divisible by 8 is 37
Check:-
The integers between 100 and 400 which are multiples of 8(divisible by 8)
104,112,120,128,136,144,152,160,168,176,184,
192,200,208,216,224,232,240,248,256, 264, 272, 280, 288, 296, 304, 312, 320, 328,336,344,352,360,368,376,384,392.
Total number of integers = 37
Verified the given relations in the given problem.
Used formulae:-
- The nth term of an AP =an = a+(n-1)d
- a = First term
- d = Common difference
- n = Number of terms
- If n AM's between two numbers a and b in an AP then d = (b-a)/(n+1)
- a = first number
- b = last number