Question

How many pairs of positive integer (a, b) exist such that HCF * (a, b) + LCM * (a, b) = 44 if a <b? ​

Answer 1

Given :  HCF (a, b) + LCM   (a, b) = 44

To Find  : How many pairs of positive integer (a, b) exist such that  a <b

Solution:

HCF (a, b) + LCM   (a, b) = 44

LCM   (a, b) = k HCF (a, b)

where k is positive integer

=> HCF (a, b) + k HCF (a, b)= 44

=> HCF (a, b) = 44/(k + 1)

k + 1 must be a factor of 44

Factors of 44 are  1, 2 , 4 , 11 , 22 , 44

k = 1 , 3 , 10 , 21 , 43

k = 1  =>   HCF = LCM = 22 => a , b = 22  but a < b hence not possible

k = 3 =>  HCF = 11  , LCM = 33  => a  = 11 ,  b = 33

k = 10 => HCF = 4  ,  LCM = 40 =>  a = 4 , b = 40   and a = 8  b = 20

k = 21 => HCF = 2   LCM = 42  =>  a = 2 , b = 42   and  a  = 6 , b = 14

k = 43 =>  HCF = 1    LCM = 43     Hence a = 1  , b = 43

HCF = 11  , LCM = 33

a = 11m    b = 11n

m and n are co prime ,  m < n

11 * 33 = 11m * 11n

=> m * n =  3

m = 1 and n = 3

HCF = 4, LCM = 40

a = 4m    b = 4n

m and n are co prime ,  m < n

4 * 40 = 4m * 4n

=> m * n =  10

m = 1 and n = 10   , m = 2 and n = 5

HCF = 2   LCM = 42

a = 2m  , b = 2n

m and n are co prime ,  m < n

=> 2 * 42  = 2m * 2n

=> mn = 21   ,

m = 1  , n = 21  ,  m = 3  and n = 7

Pairs of positive integers of ( a , b) are:

( 1  , 43)  , ( 2 , 42 ) , (4 , 40) , ( 6 , 14) , ( 8 , 20) , (11 , 33)

Hence 6 pairs

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