How many numbers are there between 1 and 1000 which when divided by 7 leave remainder 4?

Solution:

A number N is divided by 7 leaves a remainder 4.

∴ N = 7q + 4

N can take values 4, 11, 18, ..... 998

Now,

4, 11, 18, ..... 998 are in arithmetic progression.

First term a = 4

common difference d = 7

last term l = 998

We know that,

l = a + (n − 1)d

⇒ 998 = 4 + (n − 1)7

⇒ 998 = 4 + 7n − 7

⇒ 998 = 7n − 3

⇒ 1001 = 7n

$\Rightarrow n=\frac{1001}{7}$

$\Rightarrow n=143$

Hence, 143 numbers are there between 1 and 1000 which when divided by 7 leave remainder 4.