Expert answers to "What is the sum of number of digits..."

High School
Algebra
Pre-Algebra
Decimals

Samir Duffy

Answered question

2022-03-12

What is the sum of number of digits of the numbers

2^{2001}

and

5^{2001}

Answer & Explanation

Avery Campbell

Beginner2022-03-13Added 6 answers

Step 1
It looks like you don't need logarithms or any calculator to solve this problem. Let's start.
First, observe that the following inequalities hold:

10^{m} < \underset{m + 1 \sim digits}{\underset{⏟}{2^{2001}}} < 10^{m + 1}

10^{n} < \underset{n + 1 \sim digits}{\underset{⏟}{5^{2001}}} < 10^{n + 1}

You get,

10^{m + n} < 10^{2001} < 10^{m + n + 2}

2001 = m + n + 1

m + n = 2000

Finally, the sum of digits of

2^{2001}

and

5^{2001}

i equal:

m + 1 + n + 1 = m + n + 2

= 2000 + 2

= 2002

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