What is the greatest five-digit positive integer which when divided by

Notice that there is constant difference between each divisor and the respective remainder.
For example, when divisor is 12, remainder is 5; the difference between 12 and 5 is 7.
Similarly, when divisor is 15, remainder is 8; the difference between 15 and 8 is 7.

In such a case where a given number, say N, is divided by many divisors, say a, b, c. etc., yielding different remainders in such a way that the difference between the divisor and remainder is constant, the number N can be expressed as,

N = LCM (a, b, c…) k – (Common difference), where k is a positive integer.

Therefore, the numbers we are looking for are of the form,

N = LCM (12, 15, 21, 25, 28) k – 7

LCM (12, 15, 21, 25, 28) = 2100.

Therefore, N = 2100 k – 7; in other words, N + 7 = 2100k.

Since we need the greatest five digit number that satisfies the above constraint, we can divide 100,000 by 2100 and subtract the remainder.

When 100,000 is divided by 2100, the remainder is 1300. This means, the greatest five-digit integer which is a multiple of 2100 = 100,000 – 1300 = 98700

Therefore, N + 7 = 98700 and hence, N = 98693.

The correct answer option is B.