Multiple Magic
Think of any whole number. Each time you perform a sequence of operations on it, what do you notice about the divisors of your answer?
Age
11 to 14
Challenge level
Problem
Think of any whole number.
Double it and add five.
Double this answer and add two.
Now take away the number you first thought of.
Your answer will always be a multiple of which number?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: 3
Call your first number $x$.
Doubling and adding five gives $2x+5$.
Doubling and adding two gives $2(2x+5)+2 = 4x+12$.
Subtracting $x$ then leaves $3x+12 = 3(x+4)$ which is always a multiple of 3.