Question

Find the least five digit number which is divisible by 666.
(A) 10656
(B) 10646
(C) 10666
(D) 10626

Answer 1

Hint: This problem can easily be solved using a hit and trial method. First try to find the biggest four digit number that is a multiple of 666. If this number is known then the next multiple will be the least five digit number which is a multiple of 666 and hence our answer. Check with $5^{th}$, $10^{th}$, $20^{th}$ multiple of 666 and try to narrow the bandwidth within which the largest four digit multiple or the least five digit multiple may lie.

Complete step-by-step answer:
According to the question, the least five digit multiple of 666 is to be determined.
We will use hit and trial methods and will narrow down the range within which such numbers may lie. For doing so, first we will try to find the largest four digit multiple of 666. Let’s first multiply 666 by 10 and see what we get:
$ \Rightarrow 666 \times 10 = 6660$
As we can see, 10th multiple is 6660, which is way below 10000. So let’s find the $20^{th}$ multiple of 666, we’ll get:
$ \Rightarrow 666 \times 20 = 13320$
Clearly, the number will lie somewhere between these two. Considering $15^{th}$ multiple of 666, we’ll get:
$ \Rightarrow 666 \times 15 = 9990$
From this we have 9990, which is just 10 shy of 10000. Thus we can conclude that the $15^{th}$ multiple of 666 i.e. 9990 is the largest four digit multiple of 666.
So for the least five digit multiple of 666, we’ll consider just the next multiple of it i.e. $16^{th}$ multiple of 666. Doing this, we have:
$ \Rightarrow 666 \times 16 = 10656$
Hence 10656 is the least four digit multiple of 666.

Option (A) is the correct answer.

Note: If we reduce the number 666 in its prime factor form, we’ll get:
$ \Rightarrow 666 = 2 \times {3^2} \times 37$
From this we can conclude that any number which is a multiple of 666 will also be a multiple of $2,{\text{ }}3,{\text{ }}{3^2}$ and $37$. It will also be a multiple of combination of these factors like $2 \times 3,{\text{ }}3 \times 37,{\text{ }}2 \times 3 \times 37$ and so on.