Let your grandfather's age be x, then
![\frac{x-2}{7} =p\ .\ .\ .\ (1) \\ \\ \frac{x-3}{5} =q\ .\ .\ .\ (2) \\ \\ \frac{x-5}{11} =r\ .\ .\ .\ (3)](https://tex.z-dn.net/?f=%20%5Cfrac%7Bx-2%7D%7B7%7D%20%3Dp%5C%20.%5C%20.%5C%20.%5C%20%281%29%20%5C%5C%20%20%5C%5C%20%20%5Cfrac%7Bx-3%7D%7B5%7D%20%3Dq%5C%20.%5C%20.%5C%20.%5C%20%282%29%20%5C%5C%20%20%5C%5C%20%20%5Cfrac%7Bx-5%7D%7B11%7D%20%3Dr%5C%20.%5C%20.%5C%20.%5C%20%283%29)
for some integers: p, q and r.
Solving for x in the 3 equations we have:
![x-2=7p \\ \\ \Rightarrow x=7p+2\ .\ .\ .\ (4) \\ \\ x-3=5q \\ \\ \Rightarrow x=5q+3\ .\ .\ .\ (5) \\ \\ \\ x-5=11r \\ \\ \Rightarrow x=11r+5\ .\ .\ .\ (6)](https://tex.z-dn.net/?f=x-2%3D7p%20%5C%5C%20%20%5C%5C%20%5CRightarrow%20x%3D7p%2B2%5C%20.%5C%20.%5C%20.%5C%20%284%29%20%5C%5C%20%20%5C%5C%20x-3%3D5q%20%5C%5C%20%20%5C%5C%20%5CRightarrow%20x%3D5q%2B3%5C%20.%5C%20.%5C%20.%5C%20%285%29%20%5C%5C%20%20%5C%5C%20%20%5C%5C%20x-5%3D11r%20%5C%5C%20%20%5C%5C%20%5CRightarrow%20x%3D11r%2B5%5C%20.%5C%20.%5C%20.%5C%20%286%29)
From eqtn (4), possible values of x: 9, 16, 23, 30, 37, 44, 51, 58, 65, 72, 79, 86, 93, 100
From eqtn (5), possible values of x: 8, 13, 18, 23, 28, 33, 38, 43, 48, 53, 58, 63, 68, 73, 78, 83, 88, 93, 98
From eqtn (6), possible values of x: 16, 27, 38, 49, 60, 71, 82, 93, 104
Notice that the only number that appeared in all three is 93.
Therefore, your grandfather's age is 93 years old.