If a positive integer F divides a certain positive integer M, 6 is the remainder. When F divides M+33, 4 is the remainder. Find all possible values of F.
I found 1 possible value for F, which is 7. In this case, M could equal 48. The only method I know is using guess and check, but it is not efficient and could leave out other possibilities. Please help me find all the other possible values for F. Also, how would you prove that there are no other possibilities besides these?
Thanks :)
The first divisibility condition gives $$M=kF+6$$ for some integer $k$. Since the remainder is always less than the divisor so one must also have $6<F$. Similarly, the second divisibility condition gives $$M+33=sF+4\Rightarrow M=sF-29$$ with $4<F$ for some integer $s$. So $$kF+6=sF-29$$ This gives $$F(s-k)=35$$ with $F>6$ and therefore you get the possible values of $F$ as $7$ and $35$.