In successive division quotient obtained from the previous division is divided by the next divisor. For example if 52 is divided by 3, 4 and 7 successively, corresponding remainders will be 1, 1 and 4 as explained below:

Now let us look at the question from a different angle. Suppose we have to find a number which when divided by 3, 4 and 7 successively the corresponding remainders are 1, 1 and 4. We start the solution backward. Suppose the last quotient is x, that is obtained after dividing the previous dividend by 7, hence previous dividend must be \(7x + 4\). Proceeding in the same manner, we get the number as \(84x + 52\).

Smallest such number is obtained by putting \(x\) as 0, number is 52. Second smallest number is obtained by putting \(x = 1\), number = 136

Example 28: Find the smallest four digit number which when divided by 3, 4, 5 and 6 successively, the corresponding remainders are 1, 2, 3 and 4 respectively. Also find the remainder when this number is divided by 60.

Solution: Proceeding in the same manner as discussed above, we get the following table,

Hence the number is \(360x + 283\), smallest 4 digit number is obtained by putting \(x = 2\), hence the number is \(360 × 2 + 283 =1003\).

Remainder when this number is divided by 60 =\(\frac{{360x + 283}}{{60}}= 43\).