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# If 10^50-74 is written as an integer in base 10 notation

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Manager
Joined: 06 Apr 2010
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If 10^50-74 is written as an integer in base 10 notation  [#permalink]

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26 Aug 2010, 07:42
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37
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Difficulty:

45% (medium)

Question Stats:

66% (01:51) correct 34% (02:06) wrong based on 759 sessions

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If 10^50-74 is written as an integer in base 10 notation, what is the sum of the digits in that integer?

A. 424
B. 433
C. 440
D. 449
E. 467
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Joined: 02 Sep 2009
Posts: 50058

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26 Aug 2010, 08:25
12
9
udaymathapati wrote:
If 10^{50}-74 is written as an integer in base 10 notation, what is the sum of the digits in
that integer?
A. 424
B. 433
C. 440
D. 449
E. 467

$$10^{50}$$ has 51 digits (1 followed by 50 zeros). $$10^{50}-74$$ has 50 digits: last 2 digits are 2 and 6 (26) and first 48 digits are 9's.

Like 1,000-74=926.

So the sum of the digits is $$9*48+2+6=440$$.

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Manager
Joined: 24 Dec 2009
Posts: 184

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26 Aug 2010, 23:12
C for me too..Excellent expln by Bunnel. Thanks.
Retired Moderator
Joined: 20 Dec 2010
Posts: 1835

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19 Feb 2011, 14:03
The integer is going to have 48 9's.
Last 2 digits will be 26

48*9 + 2 + 6 = 432+8 = 440.

Ans: "C"
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Re: If 10^50-74 is written as an integer in base 10 notation  [#permalink]

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23 Nov 2013, 12:05
I don't understand why in the question it is mentioned "in base 10 notation"

Maybe its because English is not my mother tongue but that instruction really confused me. I thought I was looking for a number like "ten to the power of something".
Math Expert
Joined: 02 Sep 2009
Posts: 50058
Re: If 10^50-74 is written as an integer in base 10 notation  [#permalink]

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23 Nov 2013, 13:07
1
1
Lobro wrote:
I don't understand why in the question it is mentioned "in base 10 notation"

Maybe its because English is not my mother tongue but that instruction really confused me. I thought I was looking for a number like "ten to the power of something".

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.

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if-10-50-74-is-written-as-an-integer-in-base-10-notation-51062.html

Hope this helps.
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Re: If 10^50-74 is written as an integer in base 10 notation  [#permalink]

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04 Apr 2014, 01:26
100 - 74 = 26

The last 2 digits of the term would be 26; all else would be 9

99999999......26

Important rule:

Sum of ANY NUMBER added to 9 would give the SAME value of itself

For example; Consider number = 13

Sum of digits = 1+3 = 4

Adding 9 to 13 = 22 = 2+2 = 4

So the sum would always remain the same;

Back to our problem

99999999......26 = The sum of this number will add up to 2+6 = 8

From the options available, A & B can be discarded

9x1 = 9
9x2= 18
9x3= 27
9x4= 36
9x5= 45
9x6= 54
9x7= 63
9x8= 72 ........................................ 48th time
9x9= 81
9x10=90

99999999......26

$$10^{50}- 74$$ means 9 would be repeated 48 times; so last digit would be 2

Now we have 2+2+6 = 10 (Last digit is 0)

Only option C best fits = 440

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Posts: 956
Re: If 10^50-74 is written as an integer in base 10 notation  [#permalink]

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01 Apr 2018, 05:00
Bunuel wrote:
udaymathapati wrote:
If 10^{50}-74 is written as an integer in base 10 notation, what is the sum of the digits in
that integer?
A. 424
B. 433
C. 440
D. 449
E. 467

$$10^{50}$$ has 51 digits (1 followed by 50 zeros). $$10^{50}-74$$ has 50 digits: last 2 digits are 2 and 6 (26) and first 48 digits are 9's.

Like 1,000-74=926.

So the sum of the digits is $$9*48+2+6=440$$.

Hello Bunuel why is the same question tagged both as 600 and 700 level question

here the same https://gmatclub.com/forum/if-10-50-74- ... 51062.html

is it 600 and 700 level question ?
Director
Joined: 09 Mar 2016
Posts: 956
Re: If 10^50-74 is written as an integer in base 10 notation  [#permalink]

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01 Apr 2018, 05:18
Bunuel wrote:
udaymathapati wrote:
If 10^{50}-74 is written as an integer in base 10 notation, what is the sum of the digits in
that integer?
A. 424
B. 433
C. 440
D. 449
E. 467

$$10^{50}$$ has 51 digits (1 followed by 50 zeros). $$10^{50}-74$$ has 50 digits: last 2 digits are 2 and 6 (26) and first 48 digits are 9's.

Like 1,000-74=926.

So the sum of the digits is $$9*48+2+6=440$$.

generis can you please explain ?

i dont understand how after $$10^{50}-74$$ we have 50 digits

And how we get "last 2 digits are 2 and 6 (26) and first 48 digits are 9's"
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Joined: 26 Mar 2013
Posts: 1842
Re: If 10^50-74 is written as an integer in base 10 notation  [#permalink]

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01 Apr 2018, 05:25
1
dave13 wrote:
Bunuel wrote:
udaymathapati wrote:
If 10^{50}-74 is written as an integer in base 10 notation, what is the sum of the digits in
that integer?
A. 424
B. 433
C. 440
D. 449
E. 467

$$10^{50}$$ has 51 digits (1 followed by 50 zeros). $$10^{50}-74$$ has 50 digits: last 2 digits are 2 and 6 (26) and first 48 digits are 9's.

Like 1,000-74=926.

So the sum of the digits is $$9*48+2+6=440$$.

Hello Bunuel why is the same question tagged both as 600 and 700 level question

here the same https://gmatclub.com/forum/if-10-50-74- ... 51062.html

is it 600 and 700 level question ?

Hi dave13

Have you posted question in GMATclub before? When someone posts a question, s/he chooses the level by checking. So it is subjective to the person who solves the question. I can say easy but another considers it hard or medium. Personally, I trust he level of a question when an instructor like Bunuel...etc tags the level.

Hi Bunuel
Can you merge the same question in one post?
https://gmatclub.com/forum/if-10-50-74- ... 51062.html
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Joined: 22 May 2016
Posts: 2040
If 10^50-74 is written as an integer in base 10 notation  [#permalink]

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01 Apr 2018, 12:57
1
dave13 wrote:
Bunuel wrote:
udaymathapati wrote:
If 10^{50}-74 is written as an integer in base 10 notation, what is the sum of the digits in
that integer?
A. 424
B. 433
C. 440
D. 449
E. 467

$$10^{50}$$ has 51 digits (1 followed by 50 zeros). $$10^{50}-74$$ has 50 digits: last 2 digits are 2 and 6 (26) and first 48 digits are 9's.

Like 1,000-74=926.

So the sum of the digits is $$9*48+2+6=440$$.

generis can you please explain ?

i dont understand how after $$10^{50}-74$$ we have 50 digits

And how we get "last 2 digits are 2 and 6 (26) and first 48 digits are 9's"

dave13 , I've seen you use patterns. Good instinct. Use a pattern. (I think you missed the "1,000" pattern above.)

First we have to figure out what the digits ARE. That's just subtraction. Start with 100. (You could start with 1,000, which would be a little more accurate. 1,000 - 74 = 926. There is a 9. But, see below, 26 is always there.)

Given (100-74), what is the sum of the digits?*
100-74 = 26. Sum of the digits? (2+6)=8

How many digits in the answer? TWO. You wrote: "i dont understand how after $$10^{50}-74$$ we have 50 digits"

The exponent, 50, gives us a clue. Back to the earlier pattern.
100 = 10$$^2$$. How many digits in $$10^2 -74?$$ TWO digits in the answer, 26

But we have to be careful. If subtracting a positive integer (less than 100) from 10$$^2$$, the possible number of digits in the answer is two OR one.
Two digits: (100-74) = 26
One digit: (100-94) = 6

The exponent is a clue only. Simple subtraction, with a few examples, will tell us how many digits. So let's go higher by powers of 10:
10$$^3$$ = 1,000
10$$^4$$ = 10,000
10$$^5$$ = 100,000

Subtract 74 from each one. (Writing on paper really shows the pattern. Formatting here is hard):
(1,000 - 76) = 926
(10,000-76) = 9,926
100,000-76 = 99,926

$$(10^3 - 74)$$ has THREE digits. One 9, and 26
$$(10^4 - 74)$$ has FOUR digits. Two 9s, and 26
$$(10^5 - 74)$$ has FIVE digits. Three 9s, and 26

1) We are getting the same number of digits as the exponent on 10
2) The last two digits will always be 26
3) We have to borrow to move the initial 1 to the hundreds place. So there are repeated 9s. And only 9s until 26.
4) How many 9s? Exactly TWO fewer than 10's exponent (because 2 and 6 "use up" two of the digits)

Finally, SUM of the digits?
Back to the pattern
(1,000 - 76) = 926
(10,000-76) = 9,926
100,000-76 = 99,926
$$10^3 - 74 = ((1*9)+26)=(9+26)=35$$
$$10^4-74 = ((2*9)+26)) =(18+26)=44$$
$$10^5-74= ((3*9+26))=(27+26)=53$$

Try extrapolating from the pattern above for
What is the sum of the digits of $$10^{50} - 74$$?

*A fancy way to ask that question: If $$10^{2} - 74$$ is written as an integer in base 10 notation, what is the sum of the digits in that integer?

Does that help?
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