Show that number is divided by $137$?

Question

I want to show that if we are given a $4$-digit number and "double" it, the $8$-digit number that we become is divided by $137$.

How can we show this? Could you give me a hint?

We have to find what the number that we get is equal to modulo $137$ . But how can we get such an info?

Answer 1

Take any number: ( $n,m,s,t \in \{0,1,...,9\}, n\neq 0$ )

We form 4-digit number : W = $1000n+100m+10s+t $

Now, we "double" it, getting:

$1000\cdot10000n + 100\cdot10000m + 10\cdot10000s + 10000t + 1000n+100m+10s+t$

We can take $(1000n+100m+10s+t)(10000 + 1) $ = $10001\cdot W$

The question is, whether $10001$ is divisible by $137$.

It clearly is, because $10001 = 137\cdot73$

So

$137 \ | \ 10001\cdot W$

Answer 2

Starting out: We need to precisely define our question. This means two things:

• Formulate what "doubling" means in terms of arithmetic operations
• Figure out precisely what we need to compute about the doubled number.

Process: We would like to show that for any four-digit integer $c$, the following is true: $$ \operatorname{Double}(c) \equiv 0\ \operatorname{mod} 137 $$ So we need to figure out what "$\operatorname{Double}$" means. Let's use the example of $1234$. The double of $1234$ is $12341234$. Taking it apart: $$ 12341234 = 12340000 + 1234 = 1234(10001). $$ So for a four-digit integer $c$, the double of $c$ is $10001c$.

Putting it all together: I'll let you take it from here!