Question & Answer
QUESTION

Find the smallest number of six digits which is exactly divisible by 111.

ANSWER Verified Verified
Hint: To find the smallest number of six digits which is exactly divisible by 111, we use the formula given by –
Smallest six-digit number + (111 – remainder when smallest six-digit number is divided by 111)
This would give us the required answer for the above problem.

Complete step-by-step answer:
Generally, to attempt problems involving finding the smallest number divisible by a particular number, we start with small cases and then build on to the actual problem. In this problem, we have to find the smallest number of six digits which is exactly divisible by 111. Thus, we first start by finding the smallest number of three digits which is exactly divisible by 111. We can clearly see that for this case, the number required is 111 itself. Now, we try to find a distinctive pattern based on this small case. Thus, we first start by dividing 100 (smallest three-digit number) by 111. We would see that the quotient is 0 and the remainder is 100. We now do a similar step to find the smallest number of four digits which is exactly divisible by 111. We again divide 1000 (smallest four-digit number) by 111. We get the quotient as 9 and the remainder as 1. Also, we know that the smallest four-digit number is 1110. Thus, we see that the required number always follows this formula given by the smallest n-digit number + (111 – remainder when smallest n-digit number is divided by 111). For example, in case of four-digit number, we have –
1000 + (111 – 1) = 1110
Thus, coming back to the actual problem, we have,
100000 + (111 – 100) = 100011
Hence, the correct answer is 100011.

Note: An alternative solution to finding the answer is by finding the quotient (say q) when the smallest n-digit number is divided by 111 and then to get the required number, we multiply 111 by (q+1) to get the required answer. Thus, in our case, when we divided 100000 by 111, we got the quotient as 900. Thus, we multiply 111 by 901, we would get 100011, which is the required answer.