Question

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

Answer 1

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.