Question

Determine the smallest 3-digit number that is exactly divisible by 6, 8, 10, and 12.

Answer 1

Hint:The number can be found by computing the least common multiple of 6, 8, 10 and 12. Now check if it is a 3-digit number else go for the next multiple.

Complete step-by-step answer:
Our required number will be exactly divisible by 6, 8, 10 and 12. Intuitively thinking the number will be a common multiple of 6, 8, 10 and 12. As the required number is smallest then the common multiple needs to be found is the least one. Therefore, in other words we have to find the least common multiple of them.
Now, 12 is a multiple of 6. So every multiple of 12 will be a multiple of 6 too. So 6 is not needed in order to get the least common multiple.
Now,
\[8={{2}^{3}}\] and \[10=2\cdot 5\] and \[12={{2}^{2}}\cdot 3\]
So, the least common multiple of 8, 10, 12 is = \[{{2}^{3}}\cdot 5\cdot 3\] = 120.
Again 120 is also a 3-digit number.
Hence, 120 is the required smallest 3-digit number, which is exactly divisible by 6, 8, 10, and 12.
Therefore, the required number is 120.

Note: Without finding the least common multiple by algebraic process, you can also just take the smallest number and check for which multiple of this rest of the numbers also divides it. The first such number you get will be the LCM.