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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?
        7

 
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.