Divisibility and Primes Advanced Strategy

[AsadA969]

Source: Manhattan GMAT, 4th edition!, Chapter 10, Page# 128.

Question: if a/b yields a remainder of 5, c/d yields a remainder of 8, and a, b, c and d are all integers, what is the smallest possible value for b+d?

if a=17, b=6, then remainder is 5
also, if a=26, d=9, then remainder is 8.
Calculation:
Here, b=6 and d=9. So, b+d=6+9,
So, b+d=15 (possible lowest value of b+d according to Manhattan GMAT solution)
Again,
Here, b=6 means b>5 (as b is an integer)
d=9 means d>8 (since d is an integer)
Now, adding the two inequalities we get:
b>5 + d>8
b+d>13 (which is exactly b+d=14)
Why do we not consider 14 as the lowest possible values of b+d?
I’ll be very glad if I get the excellent response as I always expect it from you.
Good luck Ron for every cases of your life!
Thanks…

[tim]

It's a little hard to tell where the problem you're quoting ends and where your commentary begins, but I'm guessing it's somewhere around where you said b=6 means b>5. Following that logic I could also say that b>0 and d>0 so b+d>0. THIS IS TRUE, but it doesn't give the answer to the problem. Just because I've proven that b+d>0 doesn't mean that the answer is 1. You have to find the smallest value that *works*, not just the smallest value that fits some over-inclusive fact. A better way to look at this would be to say that b>=6 and d>=9, giving us b+d>=15 which actually works.