Question

Find the least number of five digits that is exactly divisible by 16, 18, 24 and 30.

Answer 1

LCM of 16, 18, 24 and 30:

$\begin{array}{l}2\overline{)16,18,24,30}\\ 2\overline{)8,9,12,15}\\ 2\overline{)4,9,6,15}\\ 2\overline{)2,9,3,15}\\ 3\overline{)1,9,3,15}\\ 3\overline{)1,3,1,5}\\ 5\overline{)1,1,1,5}\\ \text{1,1,1,1}\end{array}$

LCM = 2⁴ × 3² × 5 = 720

We have to find the least five-digit number that is exactly divisible by 16, 18, 24 and 30.
But LCM=720 is a three digit number.

The least five digit number = 10000
Dividing 10000 by 720, we get:
$\begin{array}{l}720\begin{array}{c}13\\ \overline{)10000}\end{array}\\ \frac{\text{-}720}{\text{28}00}\\ \frac{-2160}{640}\\ \\ \text{The greatest four-digit number exactly divisible by 720 = 10000-640}\\ \text{=9360}\\ \text{So, the least five-digit number exactly divisible by 720 = 9360 + 720}\\ \text{= 10080}\end{array}$