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What is the sum of all digits for the number 10^30 - 37 ? A

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by Brent@GMATPrepNow » Thu Oct 06, 2016 11:38 am
alanforde800Maximus wrote:What is the sum of all digits for the number 10^30 - 37 ?

A. 63
B. 252
C. 261
D. 270
E. 337
10^30 = 1000000...thirty 0's in total....000000
In other words, 10^30 is a 31-digit number.

So, for example, 10^30 - 1 is a 30-digit number.
In fact, 10^30 - 1 = 9999...thirty 9's in total...99999

Likewise, 10^30 - 37 is a 30-digit number.
Since 100 - 37 = 63, we know that 10^30 - 37 = 9999999.....9999963
Since 10^30 - 37 is a 30-digit number, we know that this value has 28 nines followed by 63
That is 100 - 37 = 999,999,999,999,999,999,999,999,999,963

The sum of all digits = (28)(9) + 6 + 3
= 261
= C

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by GMATGuruNY » Thu Oct 06, 2016 11:38 am
alanforde800Maximus wrote:What is the sum of all digits for the number 10^30 - 37 ?

A. 63
B. 252
C. 261
D. 270
E. 337
Test easy cases and look for a pattern.

10³ - 37 = 963.
10� - 37 = 9963.
10� - 37 = 99963.

In every case:
The number of 9's is equal to TWO LESS THAN THE EXPONENT.
The last two digits are 6 and 3.
Thus, 10³� - 37 will be composed of twenty-eight 9's, one 6 and one 3, yielding the following sum:
(9*28) + 6 + 3 = 252 + 9 = 261.

The correct answer is C.
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by crackverbal » Thu Oct 06, 2016 11:18 pm
Hi Alanforde800Maximus,

Just say for an example if the question was asked here instead of 10^30 - 37 as 10^3 - 37,

Then sum of the digits would be, 1000- 37 = 963, where the sum of the digits are 9+6+3.

That is 9 + 9

So for 1000 = 10^3, we had two 9's (that is one less than the number of zeros)

Similarly then for 10^30, we should have twenty nine 9's

So sum of the digits will be 29 * 9 = 261,

Also, here you no need to multiply 29* 9, here units place when 9 is multiplied by 9 will be one, so only
answer which ends with one is answer choice is C.

Remember sometimes for topics like Exponent roots, Probability and Counting methods etc when a GMAT question
is given for a bigger number, try to solve using a smaller number from that you can figure out a pattern to answer the question.

Hope this helps �
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