There are two parts of this Question:

Part 1- Divided by 2, 3, 4, 5 and 6 leaves the remainders 1, 2, 3, 4, 5 respectively.

Part 2 - Least multiple of 7

SOLUTION:

If the difference between the divisor and the remainder is SAME, then there is aneasyway of solving the remainder questions.

Part 1:

divisor is 2 and remainder is 1. Difference is 1 (2–1)

divisor is 3 and remainder is 2. Difference is 1 (3-2)

and so on, you can see that the difference is 10 in all the cases.

Number Would be:

LCM of divisors - difference ,OR, LCM (2, 3, 4, 5, 6) - 1 = 60 - 1 = 59

Next numbers satisfying the property in part 1 will be obtained by adding LCM to 59. So next number is 119, 179 and so on.General format of these numbers is 60K + 59.

Part 2:

Finding a number of the format60K + 59 which is divisible by 7.

Put K = 0, number is 59 which is NOT divisible by 7.

Put K = 1, number is 119 which is divisible by 7.

Hence answer is 119.

NOTE - You might have observed it while solving part 1 only that 119 is divisible by 7. Such an elaborated Part 2 solution is required only if you are NOT able toobservethe answer quickly.