Question

Find LCM by division method: $48,72,80$.

Answer 1

Hint: To find the least common multiple of three numbers using the division method, we write the given numbers in a row separated by commas. Then we first divide the numbers by the least common prime number. We stop dividing the numbers when all the numbers are brought down to their lowest form i.e. they are broken down into prime numbers. The required LCM then becomes the product of common and uncommon prime factors of the given numbers.

Complete step-by-step answer:
Here, we are given the numbers $48,72,80$.
We need to find their least common multiple(LCM) using the division method.
So we start by writing all the numbers in a row, each of them separated by a comma.
Then we find the least prime number that divides the numbers.
So we divide all the numbers by the smallest prime number which is $2$ over here.
Then we put the quotient directly below the numbers and the ones which do not get divided remain the same.
But here all of the numbers get divided by $2$.
After that, we again divide by $2$, and all the numbers get divided. So we divide the numbers by $2$ a total of four times. Then we divide by 3, and two numbers get divided. Again on dividing by 3, one number gets divided. Lastly, dividing by 5, only one number gets divided. Therefore the common and uncommon prime numbers that are used to find the LCM by division method are $2,2,2,2,3,3,5$.
$
2\left| \!{\underline {\,
{48,72,80} \,}} \right. \\
2\left| \!{\underline {\,
{24,36,40} \,}} \right. \\
2\left| \!{\underline {\,
{12,18,20} \,}} \right. \\
2\left| \!{\underline {\,
{6,9,10} \,}} \right. \\
3\left| \!{\underline {\,
{3,9,5} \,}} \right. \\
3\left| \!{\underline {\,
{1,3,5} \,}} \right. \\
5\left| \!{\underline {\,
{1,3,5} \,}} \right. \\
0\left| \!{\underline {\,
{1,1,1} \,}} \right. \\
$
Now, we multiply the above-mentioned prime numbers to get the required least common multiple.
$\therefore $ The required LCM is
$
= 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5 \\
= {2^4} \times {3^2} \times 5 \\
= 16 \times 9 \times 5 \\
= 720 \\
$
Hence, the required LCM of $48,72,80$ by division method is $720$.

Note: In this division method, we always divide the numbers by prime numbers. One must not use composite numbers even if the numbers get divided by them. The numbers that do not get divided at any stage remain the same in the next row. To find the least common multiple we have to keep dividing until we get $1$ for all numbers in the row.