Question

If g.c.d of two numbers is 8 and their product is 384, then their l.c.m is ____. \[\]

Answer 1

Hint: We recall the definitions of the greatest common divisor (gcd) and least common multiple. We use the property that the product of gcd and lcm of two numbers is the product of the numbers itself. We divide the product by the gcd the two numbers to get their lcm. \[\]

Complete step by step answer:
We know that in the arithmetic operation of division the number we are going to divide is called the dividend, the number by which divides the dividend is called the divisor. We get a quotient which is the number of times the divisor is of dividend and also remainder obtained at the end of the division. If the number is $ n $ , the divisor is $ d $ , the quotient is $ q $ and the remainder is $ r $ , then by Euclid’s lemma we have
\[n=dq+r\]
If the remainder $ r=0 $ we call the number $ n $ a multiple of $ d $ and $ q $ . We also say $ d,q $ exactly divide $ n $ and call $ d,q $ the factors of $ n $ .
If two numbers say $ m $ and $ n $ are exactly divided by some divisor $ d $ then $ d $ is called a common factor of $ m,n $ . If $ d $ is the largest of the common factors then $ d $ is called greatest common divisor (gcd) or highest common factor(hcf) of the two numbers. We denote
\[d=\gcd \left( m.n \right)\]
If two numbers $ m $ and $ n $ exactly divide a number say $ r $ , then $ r $ is called a common multiple of $ m $ and $ n $ . If $ r $ is the smallest of all common multiples then we say , $ r $ is the least common multiple of $ m,n $ . We denote
\[r=\operatorname{lcm}\left( m,n \right)\]
We know that the product of gcd and lcm of two numbers is the product of numbers themselves. It means
\[\gcd \left( m,n \right)\times \text{lcm}\left( m,n \right)=m\times n........\left( 1 \right)\]
We are given in question that the g.c.d of two numbers is 8 and their product is 384. Let us denote the two numbers as $ m $ and $ n $ . So we have $ \gcd \left( m,n \right)=8 $ and $ m\times n=384 $ . We put these values in the equation to have;
\[\begin{align}
  & \gcd \left( m,n \right)\times \text{lcm}\left( m,n \right)=m\times n \\
 & \Rightarrow 8\times \text{lcm}\left( m,n \right)=384 \\
\end{align}\]
We divide both sides of above step by 8 to have;
\[\Rightarrow \text{lcm}\left( m,n \right)=\dfrac{384}{8}=48\]
So the lcm of two numbers is 48. \[\]

Note:
We can prove the formula $ \gcd \left( m,n \right)\times \text{lcm}\left( m,n \right)=m\times n $ taking $ m=a\gcd \left( m,n \right) $ and $ n=b\gcd \left( m,n \right) $ where $ a,b $ are any non-zero integers such that they are co-primes which mean two numbers which have gcd 1. We then have $ \operatorname{lcm}\left( m,n \right)=a\times b\times \gcd \left( m,n \right) $ . We can find the gcd and lcm of two numbers by prime factorizations or tabular method.