Question

The sum of two no.s is 56 and their lcm is 105. Find their hcf

Answer 1

Heya User,

--> Let the no.s be a, b
--> hcf( a , b ) = k

=> a = km || b = kn || ( m, n ) = 1 { they are relatively prime }

Now, km + kn = 56
=> k ( m + n ) = 56

Also, lcm( a , b ) = kmn = 105

However, since hcf ( m , n ) = 1,
---> hcf ( m + n , n ) = 1
---> hcf ( m + n , mn ) = 1
=> hcf [ k(m+n) , kmn ) = k
=> hcf [ 56 , 105 ] = k
=> k = 7 

=> ( m + n ) = 8 || mn = 15
=> m = 5 , n = 3 .. or vice versa..
=> a = km = 7 * 5 = 35 || b = kn = 7 * 3 = 21

Hence, ( a , b ) = ( 35 , 21 ) DONE ....

Eh! We were to find the hcf =_= 
But HCF = k , ^ _ ^ which we saw abv ...
So, HCF [ a , b ] = k = 7