Question

What is the sum of all natural number s between 300 and500 leaves remind her 3 on division by 7?

Answer 1

Numbers that are divisible by 7 : let 7k where k is an integer.

Then numbers that give remainder 3 : 7k+3

Since the possible value of the number is between 300 and 500

We can use inequality here.

• 300<Number<500

We get $300<7k+3<500$

$297<7k<497$

$\frac{297}{7} <k<\frac{497}{7}$

By using improper fraction

$42\frac{3}{7} <k<71$

Therefore, k is all integers between 42 and 71

The number we are looking for is 7k+3

7(43+44+...+69+70)+3

=7 × 1/2 × (43+70) × 28 + 3  **⁽¹⁾

=13450

Hope this helps your studing.

• ⁽¹⁾ Search for Gauss's way of 1+2+...+99+100