How many positive integers are there from 0 to 1000 that leave a remainder of 3 on division by 7 and...........

Explanation:

Number should be of the form 7n + 3 and 4m + 2.
The LCM of 7 and 4 is 28. So, let us see what are the possible remainders when we divide this number by 28.
A number of the form 7n + 3 can be written as 28K + 3 or 28k + 10 or 28 + 17 or 28k + 24.
A number of the form 4m + 2 can be written as
28l + 2, 28l + 6, 28l + 10, 28l + 14, 28l + 18, 28l + 22, 28l + 26.
Within these, the only common term is 28K + 10.
The numbers in this sequence are 10, 38, 66.....990.
We still need to figure out how many numbers are there in this sequence.
We are going in steps of 28, so let us see if we can write these numbers in terms of 28p + r.
10 = 28 * 0 + 10
38 = 28 * 1 + 10
66 = 28 * 2 + 10
94 = 28 * 3 + 10 ..........
990 = 28 * 35 + 10
There are 36 numbers in this sequence.