The four digit smallest positive number which when divided by 4, 5, 6 or 7, it leaves always the remainder as 3:

Explanation:-

The least possible number = (LCM of 4, 5, 6 and 7) + 3
= 420 + 3 = 423
The next higher number is,
(420m + 3), now we put a least value of m such that
(420m + 3) ≥ 1000
At m = 3,
value = 420 * 3 + 3 = 1263