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What is the sum of the digits in the decimal notation of the number

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Bunuel
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What is the sum of the digits in the decimal notation of the number [#permalink] New post   07 Feb 2020, 00:49
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What is the sum of the digits in the decimal notation of the number \(10^{20} - 16\)?

A. 156
B. 158
C. 174
D. 176
E. 456

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Bunuel
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Re: What is the sum of the digits in the decimal notation of the number [#permalink] New post   07 Feb 2020, 00:50
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Bunuel wrote:
What is the sum of digits in decimal notation of number \(10^{20} - 16\)?

A. 156
B. 158
C. 174
D. 176
E. 456


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Re: What is the sum of the digits in the decimal notation of the number [#permalink] New post   10 Feb 2020, 00:30
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one alternate way
value of 10^20 will have total 21 digits i.e 20 zeroes and 1 one
and last two digits of \(10^{20} - 16\) = 84 ; sum ; 3
other 19 digits will be 9
so total sum ; 19*9+3 ; 174
IMO C


Bunuel wrote:
What is the sum of digits in decimal notation of number \(10^{20} - 16\)?

A. 156
B. 158
C. 174
D. 176
E. 456
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Re: What is the sum of the digits in the decimal notation of the number [#permalink] New post   07 Feb 2020, 02:08
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Bunuel wrote:
What is the sum of digits in decimal notation of number \(10^{20} - 16\)?

A. 156
B. 158
C. 174
D. 176
E. 456


Let's concentrate on the power (exponent) of 10 and number of 9's in digit sum

Sum of the digits in \(10^{2} - 16 = 84\) = \(8+4=12\)
Sum of the digits in \(10^{3} - 16 = 984\) = \(9+8+4=9+12\)
Sum of the digits in \(10^{4} - 16 = 9984\) = \(9+9+8+4=2*9+12\)
Sum of the digits in \(10^{5} - 16 = 99984\) = \(9+9+9+8+4=3*9+12\)
----
----
Sum of the digits in \(10^{20} - 16 = 99---984\) = \(9+9+9---9+8+4=18*9+12 = 162+12 = 174\)

Answer: Option C
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Re: What is the sum of the digits in the decimal notation of the number [#permalink] New post   07 Feb 2020, 03:30
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Bunuel wrote:
What is the sum of digits in decimal notation of number \(10^{20} - 16\)?

A. 156
B. 158
C. 174
D. 176
E. 456


Solution


First, let us check if digits of \(10^n-16\) (in decimal notion) follows any pattern or not, then we will calculate the answer.
Since 16 is greater than \(10^1\), we will start with \(10^2\)
    • \(10^2 – 16 = 100-16= 84\)
    • \(10^3 – 16 = 1000 – 16 = 984\)
    • \(10^4 – 16 = 10000 – 16 = 9984\) and so on.
We can see from the above that decimal notation of \(10^n – 16 \) follows a pattern such that, digits in decimal notion of \(10^n – 16 \) are \((n-2)\) times \(9\)s, 8 and 4.
    • Hence, the sum of digits in decimal notion of \(10^n – 16 = (n-2) *9 +8+4\)
      o The sum of digits in decimal notion of \(10^{20}-16 = (20-2) *9+8+4 = 174\)
Thus, the correct answer is Option C.
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Re: What is the sum of the digits in the decimal notation of the number [#permalink] New post   04 Jul 2023, 00:31
Expert Reply
Bunuel wrote:
What is the sum of the digits in the decimal notation of the number \(10^{20} - 16\)?

A. 156
B. 158
C. 174
D. 176
E. 456


\(10^{20}\) 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: C.
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Re: What is the sum of the digits in the decimal notation of the number [#permalink]
04 Jul 2023, 00:31
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