If x = 10^50 - 57, what is the sum of all the digits of x?

Question

(A) 390
(B) 418
(C) 420
(D) 439
(E) 449

Bunuel · Accepted Answer

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interceptor77 · Answer

Thanks for the links. For this problem I used the following reasoning. 10^50 is an eleven digit number and can be defined as 1 followed by 50 zeros, when we start to perform the subtraction of this giant number from 47 we notice the following pattern:

10^2 100-57=43 (result of two digits number)
10^3 1000-57=943 (result of three digits number)
10^4 10000-57=9943 (result of four digits number)
and so on... until 10^50

The result of the subtraction becomes a 50 digits number composed by 48 pairs of 9 followed by a 4 and 5. Therefore 48*9=432 and 432+(4+5)=439

The correct answer is D.

fireagablast · Answer

10^1 = 10
10^2 = 100
..
10^50 = 50 zeroes

1000-57 = 943 -> (3-2)9 + 4 +3 
10^50-57 = (50-2)(9) + 4 +3 = 439

Mansouri · Answer

hello  Bunuel ,

can you explain this one also. thanks man

Bunuel · Answer

10^{50}  has 51 digits: 1 followed by 50 zeros;
 10^{50}-57  has 50 digits: 48 nines and 43 at the end;

So, the sum of the digits of  10^{50}-74  equals to 48*9 + 4 + 3 = 439.

Answer: D.

Hope it helps.

Oppenheimer1945 · Answer

10^2-57=43
10^3-57=943

10^n-54=999...(n-2) times 43

sum of Digits=9(n-2)+4+3=9(48)+4+3
9(50-2)+7=450-18+7=450-11=439

AkiWho · Answer

Doesn't sum of all the digits mean digital root?
I thought we have to do 4+3+2+4+3 and so on
Why are we adding 432 + 4+3?

egmat · Answer

AkiWho  No, sum of all digits and digital root are different.

Sum of all the digits  means adding each digit exactly once and stopping there. For example, if the number is  432 , the sum of digits is  4 + 3 + 2 = 9 , and we're done. 
  Digital root  means repeatedly summing the digits until you get a single digit. For  437 : first sum  4 + 3 + 7 = 14 , then sum again  1 + 4 = 5 . The digital root is  5 .

For This Problem:

The question asks: "what is the sum of all the digits of  x ?"

This means we add each digit once only.

When  x = 10^{50} - 57 , we get:
 x = 999...999943  (48 nines, then  4 , then  3 )

Sum of digits  = 48(9) + 4 + 3 = 432 + 4 + 3 = 439

We stop here because the question asks for the sum of all digits, not the digital root.

Ishaan30 · Answer

Questions like these are a spell that urge me to say, "How will I know the answer, I'm not a calculator."
And that is when you should realize that the question is just hiding behind a pattern that, once recognized, will be easier than an egg to crack.

I start small -> 
100 - 57 = 43
1000 - 57 = 943
10,000 - 57 = 9943
100,000 - 57 = 99943

2 zeros means 0 nines.
3 zeros means 1 nine.
4 zeros means 2 nines.

Whatever the number of zeros, 2 nines less. 
50 zeros means 48 nines. 
9*48 + 4 + 3 =  439

The trick is to not panic and try to see right through the question.

If x = 10^50 - 57, what is the sum of all the digits of x?

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GMAT Problem Solving (PS) Questions

If x = 10^50 - 57, what is the sum of all the digits of x?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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My Rewards