Question:
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.