Question

Find the LCM by division method: \[24\] and \[30\]

Answer 1

Hint: Here we are asked to find the LCM of the given two numbers by division method. To find the least common multiple by the division method we need to start dividing the given numbers by a common prime number. This step will get repeated until we reach one. Then by multiplying all the prime numbers that we used to divide will give the least common multiple of the given two numbers.

Complete step by step answer:
We aim to find the least common multiple (LCM) of the given two numbers \[24\] and \[30\] using the division method.
For that we first need to write these numbers in a single row separated by a comma, let us do that.
 \[\left| \!{\underline {\,
  {24,30} \,}} \right. \]
Next, we have to find a least prime number that divides any one of the numbers without leaving any remainder. Here we have \[24\] and \[30\] , a prime number \[2\] will divide both numbers evenly. Thus, we get
 \[
  2\left| \!{\underline {\,
  {24,30} \,}} \right. \\
  {\text{ }}\left| \!{\underline {\,
  {12,15} \,}} \right. \\
 \]
Now we got \[12\] and \[15\] . We will now again find a least prime number that divides any one of these numbers. The prime number \[2\] will divide the number \[12\] and the other number will remain the same. Thus, we get
 \[
  2\left| \!{\underline {\,
  {24,30} \,}} \right. \\
  2\left| \!{\underline {\,
  {12,15} \,}} \right. \\
  {\text{ }}\left| \!{\underline {\,
  {{\text{ }}6,15} \,}} \right. \\
 \]
Now we got \[6\] and \[15\] . We have to again find a least prime number that divides any one of these numbers. The prime number \[2\] will divide the number \[6\] and the other number will remain the same. Thus, we get
 \[
  2\left| \!{\underline {\,
  {24,30} \,}} \right. \\
  2\left| \!{\underline {\,
  {12,15} \,}} \right. \\
  2\left| \!{\underline {\,
  {{\text{ }}6,15} \,}} \right. \\
  {\text{ }}\left| \!{\underline {\,
  {{\text{ }}3,15} \,}} \right. \\
 \]
Now we got \[3\] and \[15\] . We will again find a least prime number that divides at least any one of the numbers. Here \[3\] and \[15\] both are multiples of \[3\] , and three is a prime number thus we get
 \[
  2\left| \!{\underline {\,
  {24,30} \,}} \right. \\
  2\left| \!{\underline {\,
  {12,15} \,}} \right. \\
  2\left| \!{\underline {\,
  {{\text{ }}6,15} \,}} \right. \\
  3\left| \!{\underline {\,
  {{\text{ }}3,15} \,}} \right. \\
  {\text{ }}\left| \!{\underline {\,
  {{\text{ }}1,{\text{ }}5} \,}} \right. \\
 \]
Now we have \[1\] and \[5\] . We can’t proceed further with one thus we need to find a prime number for five but five itself is a prime number thus we get
 \[
  2\left| \!{\underline {\,
  {24,30} \,}} \right. \\
  2\left| \!{\underline {\,
  {12,15} \,}} \right. \\
  2\left| \!{\underline {\,
  {{\text{ }}6,15} \,}} \right. \\
  3\left| \!{\underline {\,
  {{\text{ }}3,15} \,}} \right. \\
  5\left| \!{\underline {\,
  {{\text{ }}1,{\text{ }}5} \,}} \right. \\
  {\text{ }}\left| \!{\underline {\,
  {{\text{ }}1,{\text{ 1}}} \,}} \right. \\
 \]
Thus, we reached one for both numbers, now we will find the LCM by multiplying all the prime numbers that we used to divide those numbers. The prime numbers that we used are \[2,2,2,3,5\] . Let us find the product of these prime numbers.
 \[2 \times 2 \times 2 \times 3 \times 5 = 120\]
Hence, the least common multiple (LCM) of the numbers \[24\] and \[30\] is \[120\] .

Note: In this division method we need to divide the numbers by prime numbers only; we are not supposed to divide it by any other number. At any stage, if any number didn’t get divided by the prime number, we took it to be shifted to the next row as it is.