Question

The HCF and LCM of the two number is 16 and 192 respectively. If one of the numbers is 64 the other one is
$$\begin{array}{l}\left(\text{a}\right)\text{ 48}\\ \left(\text{b}\right)\text{ 24}\\ \left(\text{c}\right)\text{ 72}\\ \left(\text{d}\right)\text{ None}\end{array}$$

Answer 1

In this we find the other by using the relation between the HCF (highest common factor) and LCM (Least common multiply) of two numbers. The relation between the HCF and LCM of two numbers is that the product of two numbers is equal to their product of HCF and LCM.

Complete step-by-step solution:
Before start solve the given problem let states what is HCF and LCM
HCF (Highest common factor) = HCF (highest common factor) of the two numbers is the largest positive number which divided both the numbers.
LCM (Least common multiply) = LCM (Least common multiply) of the two numbers is the smallest positive number which divisible by both numbers.
Let a and b be two numbers such that one of the numbers is a = 64.
$\text{Also, given that HCF}(a,b)=16\text{ and LCM}(a,b)=192$.
Now we will use the relationship between the HCF and LCM of two numbers as follows
Product of HCF and LCM of the two numbers is equal to the product of the numbers
i.e. Product of $a$and $b$= Product of $\text{HCF}$ and $\text{LCM}$ of $a$ and b.
$\therefore$ By substituting HCF as 16, LCM as 192 and the value of a = 64, we get
$\Rightarrow$ $a\times b=\text{HCF}(a,b)\times \text{LCM}(a,b)$.
$\Rightarrow$ $64b=16\times 192$
By cross multiplication, we get
$b={\displaystyle \frac{16\times 192}{64}}=48$
Hence the other number is $b=$48.
The option (a) is right.

Note: In the problem, we should remember the relation between HCF and LCM of two numbers which is Product of $a$and $b$= Product of $\text{HCF}$ and $\text{LCM}$ of $a$ and b. Also, when you find the H.C.F and L.C.M, cross-check it once, before solving further as it will lead to fewer chances of error. Try not to make any calculation mistake while solving the question as it will lead to the wrong answer.