2011/062) The sum of two natural numbers when added to their LCM, gives a total of 143.? How many such pairs of numbers exis

let the numbers be mx and my with x and y co prime

so LCM = mxy

so mx + my + mxy = 143

or m(x+y+xy) = 143

add m on both sides

m(x+y+xy+1) = 143+m

m(x+1)(y+1) = 143 +m or (x+1)(y+1) = 143/m + 1

so m is factor or 143 can be 1,11,13, 143

m = 1=> (x+1)(y+1) = 144
144 = 2 * 72( x=1 y = 71)
= 3 * 48 (2,47)
= 4 * 36(3,35)
= 6 * 24( 5,23)
= 8 * 18 (7,17)
= 9 * 16(8,15)
= 12 * 12 (11,11)
all except (11,11) are coprimes so solution

m = 11 => (x+1)(y+1) = 14 , x + 1 = 2, x = 1 and y+1 = 7 y = 6 so 11 and 66
m = 13 => (x+1)(y+1) = 12 (x=1, y = 5), (x=2, y = 3) so 13, 65 or 26,39
m = 143 => (x+1)(y+1) = 2 so no solution

so 9 solutions (1,71),(5,23), (8,15), (7,17),(3,35), (2,47) (11,66), (13,65), (26,39)