The sum of H.C.F of two number and L.C.M of two numbers is 208 and L.C.M. is 12 times the H.C.F. If one of the numbers is 48, then

Consider the following statement:

Statement I: Second number is greater than the 48.

Statement II: (Second number)2 = 212.

Which of the following statement is correct?

(a) Only Statement I

(b) Only Statement II

(c) Both Statement I and II

(d) Neither Statement I nor II

Question

The sum of H.C.F of two number and L.C.M of two numbers is 208 and L.C.M. is 12 times the H.C.F. If one of the numbers is 48, then 

 

Consider the following statement: 

Statement I: Second number is greater than the 48. 

Statement II: (Second number)2 = 212. 

 

Which of the following statement is correct? 

(a) Only Statement I 

(b) Only Statement II

(c) Both Statement I and II 

(d) Neither Statement I nor II

ONLiNE UPSC · Accepted Answer

Let the HCF of the two numbers be H, and since LCM is 12 times the HCF, LCM = 12H. According to the question, the sum of HCF and LCM is 208:
H + 12H = 208 &rArr; 13H = 208 &rArr; H = 16.
So, LCM = 12 &times; 16 = 192. 

Let the second number be x. 

Given that one of the numbers is 48 and the product of two numbers = HCF &times; LCM &rArr; 48 &times; x = 16 &times;192 &rArr; x = (16 &times; 192) / 48 = 64.
So the second number is 64.

Now evaluating the statements:
Statement I is correct as the second number is greater than 48 &rarr; 64 > 48 
Statement II is correct as square of the second number is 212 &rArr; 64&sup2; = 4096 = 212

Hence the correct option is c.

Number Theory

HCF and LCM

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