Question

find the sum of all multiples of 7 lying between 500 and 900

Answer 1

Hello,

Numbers between 500 and 900 that are divisible by 7 are 504, 511, 518, …, 889, 896.
To find how many numbers there are, we can use the AP formula
896 = 504 + (n-1)7
or, 7(n-1) = 392
or, n-1 = (392/7) = 56
or, n = 57
Thus, there are 57 numbers between 500 and 900 that are divisible by 7.

Let’s do the same for 9. Numbers between 500 and 900 that are divisible by 9 are 504, 513, …891, 900.
900 = 504 + (n-1)9
or, n = 45
Thus, there are 45 numbers between 500 and 900 divisible by 9.

To eliminate double counting, we need to eliminate numbers divisible by (7*9) = 63.
Such numbers are 504, 567,…819, 882.
882 = 504 + (n-1)63
n = 7
Thus, there are 7 numbers between 500 and 900 divisible by 63.

So, total numbers between 500 and 900 divisible by either 7 or 9 
= 57 + 45 - 7
= 95

Thank You,
Aayush