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# What is the sum of digits in decimal notation of number (10^20) - 16?

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Math Expert
Joined: 02 Sep 2009
Posts: 62619
What is the sum of digits in decimal notation of number (10^20) - 16?  [#permalink]

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06 Feb 2020, 23:49
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35% (medium)

Question Stats:

80% (01:45) correct 20% (01:30) wrong based on 44 sessions

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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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Math Expert
Joined: 02 Sep 2009
Posts: 62619
Re: What is the sum of digits in decimal notation of number (10^20) - 16?  [#permalink]

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06 Feb 2020, 23:50
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What is the sum of digits in decimal notation of number (10^20) - 16?  [#permalink]

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07 Feb 2020, 01:08
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 $$10^{20} - 16 = 99---984$$ = $$9+9+9---9+8+4=18*9+12 = 162+12 = 174$$

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Re: What is the sum of digits in decimal notation of number (10^20) - 16?  [#permalink]

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07 Feb 2020, 02:30
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 digits in decimal notation of number (10^20) - 16?  [#permalink]

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09 Feb 2020, 23:30
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
Re: What is the sum of digits in decimal notation of number (10^20) - 16?   [#permalink] 09 Feb 2020, 23:30
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