Question

Two numbers both greater than \[29\] ,have HCF \[ = 29\] and LCM \[ = 4147\] .The sum of the numbers is
\[\left( A \right){\text{ }}666\]
\[\left( B \right){\text{ }}669\]
\[\left( C \right){\text{ }}696\]
\[\left( D \right){\text{ 96}}6\]

Answer 1

Hint: Consider any two variables as the two numbers. Then by using the concept that HCF is the highest common number present in both the numbers, find the way to find the values of the two numbers. Then put all the values required in the formula LCM(x,y) \[ \times \] HCF(x,y) \[ = \] x \[ \times \] y where x and y are two numbers. Solve it further, then compare the values of x and y with the conditions given in the question. With the help of which you can easily find the value of sum of both the numbers.

Complete step by step answer:
It is given to us that there are two numbers and both these numbers are greater than twenty nine. It is also given to us that the HCF of these numbers is twenty nine and the LCM of these numbers is \[4147\] . And we have to find the sum of these numbers.
We know that the HCF of two numbers is the highest common number which is available in both the numbers. Since HCF divides both the numbers then let's say that the two numbers are x and y. It is given to us that \[29\] is the HCF of two numbers. Therefore the numbers x and y will be
x \[ = \] HCF \[ \times \] a --------- (i)
y \[ = \] HCF \[ \times \] b --------- (ii)
where a and b are not common factors.
By putting the given value of HCF in equation (i) and (ii) we get
x \[ = \] \[29\] \[ \times \] a ---------- (iii)
and y \[ = \] \[29\] \[ \times \] b ---------- (iv)
As the relationship between the HCF and LCM of two numbers is that the product of the LCM and HCF of any two numbers is equal to the product of the given numbers. Here in this question we are assuming x and y be the two numbers. Therefore the relationship between their LCM and HCF is
LCM(x,y) \[ \times \] HCF(x,y) \[ = \] x \[ \times \] y ------------ (v)
Now substitute the values in the equation (v),
\[ \Rightarrow 4147 \times 29 = 29a \times 29b\]
On shifting twenty nine from the left hand side to the right hand side we get
\[ \Rightarrow 4147 = \dfrac{{29a \times 29b}}{{29}}\]
\[29\] in the numerator and the denominator will cancel out and we get
\[ \Rightarrow 4147 = 29ab\]
On shifting \[29\] to the right hand side we get
\[ \Rightarrow \dfrac{{4147}}{{29}} = ab\]
On dividing
\[ \Rightarrow ab = 143\]
As a and b are coprime then the possible values will be \[\left( {1,143} \right)\] and \[\left( {11,13} \right)\] . Because
When if we put \[a = 1\] then \[b = 143\] and if we put \[a = 11\] then \[b = \dfrac{{143}}{{11}} = 13\]
Now substitute \[a = 1\] and \[b = 143\] in equation (iii) and (iv). By doing this we get
\[x = 29 \times 1\]
\[ \Rightarrow x = 29\]
And \[y = 29 \times 143\]
\[ \Rightarrow y = 4147\]
Therefore, in this case \[x = 29\] and \[y = 4147\]
Now we will substitute \[a = 11\] and \[b = 13\] in equations (iii) and (iv),
\[ \Rightarrow x = 29 \times 11\] that is \[x = 319\]
\[ \Rightarrow y = 29 \times 13\] that is \[y = 377\]
In this case the values of \[x = 319\] and \[y = 377\]
It is given to us that both the two numbers are greater than \[29\] .Now let us consider the value of x for the pair \[\left( {1,143} \right)\] .In this we get \[x = 29\] but according to the question it should be greater than \[29\] instead of greater than or equal to. Therefore, we are rejecting the pair of values \[\left( {29,4147} \right)\] and now we are left with only \[\left( {319,377} \right)\] and these both the values of x and y are greater than \[29\] .
Therefore, the sum of two numbers x and y will be
\[ \Rightarrow x + y = 319 + 377\]
That is
\[ \Rightarrow x + y = 696\]

So, the correct answer is “Option C”.

Note:
Keep in mind the relationship between LCM and HCF of two numbers, LCM(x,y) \[ \times \] HCF(x,y) \[ = \] x \[ \times \] y where x and y are the two numbers. Also remember that two numbers are said to be co-prime if their HCF is one. Always solve the question by keeping in mind the conditions given in the question. Also remember that HCF of two numbers is the highest common number which is available in both the numbers.