Question

Find the smallest number divisible by each one of the numbers $8,9$ and $10$

Answer 1

Hint: First, we just need to know such things about the perfect numbers, which are the numbers that obtain by multiplying any whole numbers (zero to infinity) twice, or the square of the given numbers yields a whole number like $\sqrt 4 = 2$or $4 = {2^2}$.
LCM is the least common multiple, first to find the common multiples among two or more than two numbers and then finding the least among that is the concept.

Complete step by step answer:
First, to find the least common multiple of number $8$, eight can be rewritten as $8 = 2 \times 2 \times 2$ (all the numbers are two times the three cube)
Again, take the second number $9$ and rewrite the number as $9 = 3 \times 3$
Finally, take the last number from the given question, $10$and rewritten it as $10 = 2 \times 5$(these are possible outcomes while dividing the given numbers).
Hence the numbers $8,9$ and $10$ can be written in the form of $2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 2$ (according to the original number in the LCM form)
The process of LCM is represent below;
$
  2\left| \!{\underline {\,
  {8,9,10} \,}} \right. \\
  2\left| \!{\underline {\,
  {4,9,5} \,}} \right. \\
  2\left| \!{\underline {\,
  {2,9,5} \,}} \right. \\
  2\left| \!{\underline {\,
  {1,9,5} \,}} \right. \\
  2\left| \!{\underline {\,
  {1,9,5} \,}} \right. \\
  3\left| \!{\underline {\,
  {1,3,5} \,}} \right. \\
  3\left| \!{\underline {\,
  {1,1,5} \,}} \right. \\
  5\left| \!{\underline {\,
  {1,1,1} \,}} \right. \\
 $
Here in these numbers, we can see that $2$ appearing four times which is the perfect square (by the perfect square definition, the term is in even)
Also, the number $3$ appearing two times is the perfect square.
But the number $5$ does not appear a minimum of two times (only one time in the LCM form), to get the perfect square for the sequence of LCM form we will need to multiply the number $5$.
Hence, we get, numbers $8,9$ and $10$ can be rewritten in the form of $2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5$(which is the perfect square sequence)
Hence solving these terms, we get, $2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 5 \Rightarrow 3600$
Therefore, $3600$ is the smallest number divisible by each one of the numbers $8,9$ and $10$

Note: Since the question is to find the smallest number, we used the concept of the Least common multiple.
If the question is about the Greatest number, then there is a concept called GCD, greatest common division, which is to find the common divisor and then choose which number has the greatest value.
If the $\gcd (a,b) = 1$ is known as the relatively prime, where a and b are the integers.