Question

How many three digit numbers leave a remainder 3 on division by 4?

Answer 1

1^st three-digit number which leaves the remainder 3 when divided by 4 = 103

2^nd three-digit number which leaves the remainder 3 when divided by 4 = 107

3^rd three-digit number which leaves the remainder 3 when divided by 4 = 111

…

n^th three-digit number which leaves the remainder 3 when divided by 4 = 999

So, the arithmetic sequence is 103, 107, 111, … , 999

Common difference = 107 − 103 = 4

To get the n^th three-digit number from the 1^st three-digit number, we must add the common difference (n − 1) times to the 1^st three-digit number.

∴ 999 = 103 + (n − 1) × 4

⇒ 999 − 103 = 4n − 4

⇒ 4n − 4 = 896

⇒ 4n = 900

⇒ n = 225

Thus, there are 225 three-digit numbers which leaves the remainder 3 when divided by 4.