2012/009) The product of four distinct positive integers a,b,c,d is 8!.The no. also satisfy ab+a+b=322 and bc+b+c=398 find d

add 1 to 1st relation
ab + a + b + 1= 323 or (a+1)(b+1) = 17 * 19 ..1

add 1 to 2nd relation

bc+b+ c = 398

or bc + b+ c + 1 = 399 or (b+1)(c+1) = 19 * 21 ...2

from 1 and 2 we have b+1 = 19 or b= 18 (as b + 1 is common factor and is not 1 as b is not zero)

a + 1 = 17 or a = 16
c= 20
so abc = 16 * 18 * 20

abcd = 8!

so d = 8!/(16 * 18 * 20) = 7