How many integers bet 90 and 990 are divisible by 7?
is there any short and simple way of finding the answer?
thanks
Hi rcs!
The answer is 990 / 7 - (90 - 1) / 7, where "/" denotes a division that is rounded down.
I am assuming that you would count 990 and 90 if they would have been divisible by 7 (which they are not anyway).
To explain:
(90 - 1) / 7 is the number of integers divisible by 7 below and excluding 90.
990 / 7 is the number of integers divisible by 7 below and including 990.
The difference is the total number.
Suppose you are in the middle of a road with poles alongside the road.
And suppose you want to know the number of poles from where you are to the end of the road.
To find out you count the number that you can see from where you stand to the beginning of the road.
And if you are next to a pole you do not count that one.
Subtract from the total number of poles and there you are.
First that is the floor function. It returns an integer.
There are $\displaystyle \left\lfloor {\frac{{990}}{7}} \right\rfloor=141 $ integers from 1 to 990 that are by seven.
There are $\displaystyle \left\lfloor {\frac{{90}}{7}} \right\rfloor=12 $ integers from 1 to 90 that are by seven.
Note that neither 990 nor 90 is divisible by seven.
So you book is not correct. The answer is 129.
Suggest we find the multiple of 7 just greater than 90 and just less than 990.
We will get 91 and 987. Now all the multiples of 7 between these two numbers will form an AP with first term 91, common difference 7 and last term 987.
Question now simply is to find n such that the nth term is 987.
If a is the first term, common difference d then the nth term is given by
an = a + ( n-1) d
Thus we have 91 + ( n – 1 ) x 7 = 987
OR 91 + 7n – 7 = 987
7n = 987 – 91 + 7 = 903
n = 129 Hence there are 129 integers between 90 and 990 which are divisible by 7