For finding total numbers between 1 and 1000 which when divided by 7 leave remainder 4, firstly we will make an A.P. of those numbers which when divided by 7 leave remainder 4.

First number which when divided by 7 leave remainder 4 is 4

∴ a₁ = a = 4

Next number which when divided by 7 leave remainder 4 is 11

∴ a₂ = 11

Largest three-digit number which when divided by 7 leave remainder 4 is 998

∴ a_n = 998

⇒ A.P. is 4, 11,………,998

We know, a_n = a + (n – 1)d where a is first term or a and d is common difference and n is any natural number

⇒ a₁ = 4, a₂ = 11 and a_n = 998

Common difference, d₁ = a₂ – a₁ = 11 – 4 = 7

Now, a_n = a₁ + (n – 1)d

⇒ a_n = 4 + (n – 1)7

⇒ 998 = 4 + 7n – 7

⇒ 998 = 7n – 3

⇒ 998 + 3 = 7n

⇒ 1001 = 7n

⇒ n = 143

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