I will provide two methods for this.
Method 1 (Easy way) :
USE A CALCULATOR. You will get the answer in a matter of seconds like I got.
Clearly, the answer is 072.
Method 2 (Slight harder way) :
Here, I want to find the answer without using the calculator. Let's try it.
Note:- I will be making use of Congruence Modulo and Euler's Theorem, so these are the prerequisites.
Another way to put the question is “Find the remainder when is divided by 1000”.
First of all we factorise 1000 as:
Next, we find the remainder by and seperately.
It's obvious that divides . Hence,
Now to find the remainder by 125 (or 5^3), we use Euler's Theorem. Euler's theorem is applicable in this case since .
Hence,
Since
At this point, one may apply Chinese Remainder Theorem but checking few cases will yield the answer faster.
Since the number leaves a remainder 72 when divided by 125, the last digits can be:
, or
, or
, or
, or
, or
, or
, or
We also know that the number is divisible by 8. The only number from the above list which is divisible by 8 is 72. Hence the answer is 072.
I would like to share one more thing. I calculated 2^17 in my head (since use of calculator is not allowed). How did I do that? Let's see:
Now divide this number by 8 to get 2^17.
Easy, no?
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