Question

Determine the greatest 3-digit number exactly divisible by 8, 10 and 12.

Answer 1

Hint: Find the L.C.M of 8, 10 and 12. Find its greatest multiple which is less than 1000.

Complete step-by-step answer:
We know that the smallest number exactly divisible by 8, 10 and 12 is the least common multiple (L.C.M) of 8, 10 and 12.
Therefore, we find the L.C.M of 8, 10 & 12.

L.C.M = 2 x 2 x 2 x 5 x 3
= 120.
$\therefore $The least common multiple of 8, 10 and 12 is 120.
Since, 120 is exactly divisible by 8, 10 and 12, any multiple of 120 should also be exactly divisible by 120.
Since, we need to find the greatest 3 digit number, it has to be less than 1000.
Let us assume the required number to be the ${{n}^{th}}$ multiple of 120.
Then,
$\begin{align}
  & 120n\le 1000 \\
 & n\le \dfrac{1000}{120} \\
 & n\le 8.3 \\
\end{align}$
Since, n is an integer, n should be equal to 8.
Therefore, the required number = 120n
=960.
Answer is 960.

Note: If a number x is exactly divisible by another number, say y, then any multiple of x is also exactly divisible by y. The number which is exactly divisible by a set of numbers, is called their L.C.M i.e. least common multiple.