First, find the least common multiple (LCM) of 16, 18, 20 and 25, which is 3600. Since the number leaves a remainder of 4 when divided by these numbers, it must be of the form 3600k plus 4 for some integer k. We are given that this number is perfectly divisible by 7, meaning 3600k plus 4 divided by 7 leaves no remainder. When 3600 is divided by 7, the remainder is 2, and when 4 is divided by 7, the remainder is 4, so the expression becomes 2k plus 4 must be a multiple of 7. Setting 2k plus 4 equal to 7 gives 2k equal to 3, which has no integer solution, so we try the next multiple, 14. Setting 2k plus 4 equal to 14 gives 2k equal to 10, meaning k equals 5. Substituting k equals 5 back into 3600k plus 4 gives 18000 plus 4, which equals 18004.