x is a positive integer with digits the sum of which is 170....

Question

x is a positive integer with digits the sum of which is 170, if x = 10^20 - y , where y is a two digit integer, what could be the value of y?

a) 47

b) 53

c) 55

d) 75

e) 90

Could be a different number as well , I don't really remember all the answer choices on this mock exam.

AndrewN · Answer

First of all,  Stackitup , I would correct the source, since it cannot be the multiple sources you have cited (different test prep providers). Concerning the actual question, we need only focus on what happens at the end of the number derived from the difference. Instead of taking on a gigantic number such as 10^20, we can work on a smaller scale with something like 10^4, or 10000.

10000 - 47 = 9953

The important thing to note here is that  there will be two fewer leading 9s in the value derived than there were 0s . Thus, we can extrapolate that with the original value from the question, 10^20,  there will be eighteen leading 9s in the value derived .

18 * 9 = 162

All we need to do is find two digits at the end that add 8 to this sum. Choice (A) does just that, as 5 and 3 sum to 8.

10^{20} - 47 = 99,999,999,999,999,999,953

The sum will be

9 * 18 + 5 + 3 = 170

The answer must be (A)  . There is no need to check the other options, although we could easily see that they would not work.

(B) would end in -47, so the sum would be 162 + 11, or 173;
(C) would end in -45, so the sum would be 162 + 9, or 171;
(D) would end in -25, so the sum would be 162 + 7, or 169;
(E) would end in -10, so the sum would be 162 + 1, or 163

I hope that helps. Good luck with your studies.

- Andrew

x is a positive integer with digits the sum of which is 170....

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