M16-36

Question

What is the sum of the digits in the decimal notation of the number  10^{20} - 16 ?

A. 158
B. 162
C. 165
D. 174
E. 183

Bunuel · Accepted Answer

Official Solution:

What is the sum of the digits in the decimal notation of the number  10^{20} - 16 ?

A. 158
B. 162
C. 165
D. 174
E. 183

10^{20}  has 21 digits: a 1 followed by 20 zeros.
 
 10^{20}-16  has 20 digits: eighteen 9's and 84 at the end.
 
So, the sum of the digits of  10^{20}-16  equals  18*9+8+4=174 .

Answer: D

123Avi · Answer

Just one comment or perhaps it is my ignorance but I got confused by the text    in decimal notation

They could had just written What is the sum of digits of number XXXXXXXXXXX?

Bunuel · Answer

Based 10 notation, or decimal notation, is just a way of writing a number using 10 digits: 1, 2, 3, 4, 5, 6, 7, 8, and 0 (usual way), in contrast, for example, to binary numeral system (base-2 number system) notation. Similar questions to practice: the-sum-of-all-the-digits-of-the-positive-integer-q-is-equal-126388.html the-sum-of-the-digits-of-64-279-what-is-the-141460.html what-is-the-sum-of-all-digits-for-the-number-147057.html 10-25-560-is-divisible-by-all-of-the-following-except-126300.html if-x-is-a-positive-integer-and-10-x-74-in-decimal-notation-61013.html Hope it helps.

GeetikaR · Answer

Please explain - 10^20-16 has 20 digits: 18 9's and 84 in the end

Bunuel · Answer

Take a simple example: 10^3 has 4 digits: 1,000.

10^3 - 16 has three digits: 984 - one 9 followed by 84.

Practice similar questions from the post above.

Hope it helps.

GeetikaR · Answer

yea it definitely helped. Thanks a lot.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

Purnank · Answer

10^20 - 16 = 1000.....00000(20zeros) - 16 here last 3 digits are important 100-16 = 84 and Rest all till that 100 are 9 digits, total of 18 '9' digits = 9*18 + 8 + 4 = 174. 
Answer D.

KarishmaB · Answer

Take a smaller number to see the pattern. 
 100 - 16 = 84 
 1000 - 16 = 984 
 10000 - 16 = 9984
 
 So when you have 10^n, you will have (n-2) 9s and 84 as digits. 
 
 So when the number is 10^20 - 16, you will get 18 9s and 84. Sum of digits = 18*9 + 8 + 4 = 174

Answer (D)

bondi123 · Answer

I like the solution - it’s helpful.

gemministorm · Answer

yeah, really due to that decimal notation text, I started thinking about converting...
This was simple 10^20 - 16 = 9......9984 = 18*9 + 8 + 4 = 174

ayuj2401 · Answer

I don’t quite agree with the solution. 10 raised to the power 21 has 21 zeroes, as 10 raised to the power2 has 2 zeroes

Bunuel · Answer

Please read carefully!

We do NOT have 10^21, we have 10^20. The solution says:

10^{20}  has 21 digits: a 1 followed by 20 zeros.

The above is absolutely correct!

M16-36

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