Can You Find The Hundreds Digit of This Big Number?

A Challenging Math Puzzle

An exponent indicates the number of times a number multiplies by itself. For example 2² = 2 × 2 = 4 and 3³ = 3 × 3 × 3 = 27. Does that make sense?

Today’s problem involves a very large number multiplying itself by 2011 times 🎈

Here’s a hint: 2011 = 2000 + 11 …

I recommend you pause the article, grab your pen and paper, and give this a go. When you are ready, keep reading for the solution! ✒️

Solution

By rewriting 2011 as the sum of 2000 and 11, we can apply binomial expansion to this problem

Let us write out some of the terms of the expansion and gain some insight

A key observation is that for all terms except for the last, they all have some powers of 2000, from 2000²⁰⁰⁰ to 2000¹ and this means they are all divisible by 1000.

The same cannot be said for the last term, which comes out as 11²⁰¹¹ as 2011C2011 = 1 and 2000⁰ = 1

Think about what it means if a number is divisible by 1000.

They are all divisible by 1000. And in fact, if a number is divisible by 1000, their hundreds digit is always 0. (Can you prove why?)

This means we only have to focus on the term 11²⁰¹¹ as it determines the hundreds digit of 2011²⁰¹¹.

We will then apply a similar strategy by rewriting 11 as the sum of 10 and 1.

Here’s a challenge for you: can you apply a similar reasoning to figure out the solution if you haven’t already 🍏

Let us again write out some of the terms of the expansion

Once again, we should be able to see that all terms except for the last 3 terms are divisible by 1000 this time, and they are shown in exactly the image above.

Our final job is to simply evaluate the above expression 🎁

Let us do the math

Knowledge of binomial coefficients tells us that 2011C2010 = 2011C1 and 2011C and 2011C2009 = 2011C2.

This problem has applied the same idea many times, and this is the last time we are going to do that

The hundreds digit of this number is equal to that of 1 + 11 × 510 = 5611

And that’s our answer.

How amazing 👧

What was your thought process this time? Comment down below, I am eager to know :) 💞

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