Question

I just need help starting off this question!!
If I subtract 2 from my grandfather’s age, the result is divisible by 7. If I subtract 3 from my grandfather’s age, the result is divisible by 5. If I subtract 5 from my grandfather’s age, the result is divisible by 11. How old is my grandfather?

Answer 1

Let your grandfather's age be x, then

$\frac{x-2}{7} =p\ .\ .\ .\ (1) \\ \\ \frac{x-3}{5} =q\ .\ .\ .\ (2) \\ \\ \frac{x-5}{11} =r\ .\ .\ .\ (3)$

for some integers: p, q and r.

Solving for x in the 3 equations we have:

$x-2=7p \\ \\ \Rightarrow x=7p+2\ .\ .\ .\ (4) \\ \\ x-3=5q \\ \\ \Rightarrow x=5q+3\ .\ .\ .\ (5) \\ \\ \\ x-5=11r \\ \\ \Rightarrow x=11r+5\ .\ .\ .\ (6)$

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.