Question

Find the smallest number which is a multiple of all natural numbers from $ 2\,\,to\,\,10 $ (both inclusive).

Answer 1

Hint: To find smallest number which is a multiple of numbers between $ 2\,\,and\,\,10 $ we first write all numbers between $ 2\,\,and\,\,\,10 $ and then finding their L.C.M. as we know that L.C.M. of numbers is always divisible by numbers or we can say some multiple of each number is equal to L.C.M.

Complete step by step solution:
Given numbers from $ 2\,\,to\,\,10 $ including both $ 2 $ and $ 10 $ are given as:
  $ 2,3,4,5,6,7,8,9,10 $
Also, we know that the smallest number which is a multiple of given numbers is equal to L.C.M. of given numbers.
Therefore, to calculate or find the smallest number which is a multiple of give numbers we find their L.C.M.
  $ \begin{array}{*{20}{l}}
  2&\hline& {2 - 3 - 4 - 5 - 6 - 7 - 8 - 9 - 10} \\
\hline
  2&\hline& {1 - 3 - 2 - 5 - 3 - 7 - 4 - 9 - 5} \\
\hline
  2&\hline& {1 - 3 - 1 - 5 - 3 - 7 - 2 - 9 - 5} \\
\hline
  2&\hline& {1 - 3 - 1 - 5 - 3 - 7 - 1 - 9 - 5} \\
\hline
  3&\hline& {1 - 1 - 1 - 5 - 1 - 7 - 1 - 3 - 5} \\
\hline
  5&\hline& {1 - 1 - 1 - 5 - 1 - 7 - 1 - 1 - 5} \\
\hline
  7&\hline& {1 - 1 - 1 - 1 - 1 - 7 - 1 - 1 - 1} \\
\hline
  {}&\hline& {1 - 1 - 1 - 1 - 1 - 1 - 1 - 1 - 1}
\end{array} $
$ \Rightarrow$ L.C.M. of given numbers is = $ 2 \times 2 \times 2 \times 3 \times 3 \times 5 \times 7 $
$ \Rightarrow$ L.C.M. of given numbers is = $ 2520 $
Therefore, from above we see that the smallest number which is a multiple of numbers $ 2,3,4,5,6,7,8,9,10 $ is $ 2520. $
So, the correct answer is “ 2520”.

Note: L.C.M. means least common multiple, hence to find the smallest number which is a multiple of given numbers we always find their LCM as LCM and all its multiple are divisible by given numbers. But for smallest we always take L.C.M. as a result.