# Thread: Find the original number

1. ## Find the original number

Dr Jenson writes a number on the whiteboard. He then writes 221 to the right hand side of this number, which ends in a new number which is a multiple of the old number (without the 221). What is the original number - find all possible solutions.

I don't know if this fits under algebra, but thanks to anyone that helps.

2. Hello, BG5965!

Dr. Jenson writes a number on the whiteboard.
He then writes 221 to the right hand side of this number,
which ends in a new number which is a multiple of the old number.
What is the original number? -- Find all possible solutions.

Let $\displaystyle N$ be the original number.

If 221 is appended to the right end, the new number is: .$\displaystyle 1000N + 221$

If this is a multiple of $\displaystyle N$, we have: .$\displaystyle 1000N + 221 \:=\:k\!\cdot\!N$ . for some integer $\displaystyle k.$

Then we have: .$\displaystyle N \;=\;\frac{221}{k-1000}$

Since $\displaystyle N$ is an integer, $\displaystyle k-1000$ must be a factor of 221.

The only factors of 221 are: .$\displaystyle 1, 13, 17, 221$

Therefore: .$\displaystyle \begin{array}{ccccccc} k - 1000 \:=\: 1 & \Rightarrow & k \:=\: 1001 & \Rightarrow & \boxed{N \:=\:221} \\ k-1000 \:=\: 13 & \Rightarrow & k \:=\: 1013 & \Rightarrow & \boxed{N \:=\:17}\\ k-1000 \:=\: 17 & \Rightarrow & k \:=\: 1017 & \Rightarrow & \boxed{N \:=\:13}\\ k-1000 \:=\: 221 & \Rightarrow & k \:=\: 1221 & \Rightarrow & \boxed{N \:=\:1} \end{array}$