Question

Find the smallest four digit number which is exactly divisible by 10,12 and 14

Answer 1

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I’ll go through a step by step, as detailed as possible path on solving this problem, and the algorithm can be applied in any similar case.

• Let’s do prime factorization for the three numbers to find the least common multiple (LCM) for the given numbers:

8 = 2 * 2 * 2

10 = 2 * 5

12 = 2 * 2 * 3

• As far as 3 and 5 are odd numbers, we’ll have the following to construct the LCM for the three numbers:

2*2*2, the number’s divisible by 8

2*2*2*5 will make the number divisible by 10 (LCM for 8 and 10)

2*2*2*5*3 the number’s now divisible by 12 (LCM for 8, 10, 12)

So we have 2*2*2*3*5 = 120 and it’s the LCM of 8, 10 and 12.

• The smallest four digit number is 1000, right? :)

So now we have to find the smallest four digit number that is a multiple of 120, let’s go:

a)

1000 / 120 = 8.33

8 < 8.333 < 9

8 * 120 = 960 (it is not a four digit number)

9 * 120 = 1080 (or 960 +120) which is a four digit number and it’s the smallest four digit number which is divisible by 8, 10 and 12.

OR

b)

1000 / 120 = 8 (remainder = 40)

1000 + 120 - 40 = 1080

Thus, 1080 is the least common four digit multiple of 8, 10 and 12 (smallest 4 digit number that is exactly divisible by 8, 10 and 12 like you have stated in the question).