Answer: a multiple of $7$.

Using letters

Let the number thought of be $x$

Double it, then add 3:         $2x+3$

Multiply your answer by 4: $4(2x+3)=8x+12$

And then take away 5:       $(8x+12)-5=8x+7$

Now take away the number you first thought of: $(8x+7)-x=7x+7 = 7(x+1)$ multiple of $7$.

Using a diagram

Think of a whole number            

double it, then add 3                   

Multiply your answer by 4 and then take away 5                 

Now take away the number you first thought of                     

Your answer will always be a multiple of which number?