The sum of the digits of integer z is 186 and z=10^n -4. What is the value of positive integer n?
(A) 19
(B) 20
(C) 21
(D) 22
(E) 23
[spoiler]OA=C[/spoiler]
Source: Veritas Prep
The sum of the digits of integer z is 186 and
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$$Z=10^n-4$$
$$When\ n=1,\ z=6$$
$$When\ n=2,\ z=96$$
$$When\ n=3,\ z=996$$
$$When\ n=4,\ z=9996$$
$$When\ n=k,\ z=10^n-4\ will\ have\ \left(k-1\right)\cdot9\ and\ the\ last\ digit\ will\ always\ ne\ 6$$
$$Therefore,\ 186-6=180$$
$$and\ \frac{180}{9}=20$$
$$Hence,\ k-1=20$$
$$k=20+1=21\ \ \ \ \ \ OPTION\ C$$
$$When\ n=1,\ z=6$$
$$When\ n=2,\ z=96$$
$$When\ n=3,\ z=996$$
$$When\ n=4,\ z=9996$$
$$When\ n=k,\ z=10^n-4\ will\ have\ \left(k-1\right)\cdot9\ and\ the\ last\ digit\ will\ always\ ne\ 6$$
$$Therefore,\ 186-6=180$$
$$and\ \frac{180}{9}=20$$
$$Hence,\ k-1=20$$
$$k=20+1=21\ \ \ \ \ \ OPTION\ C$$
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If z = 10^n - 4, then z = 99...996 where there are (n - 1) nines (for example, if n = 3, z = 10^3 - 4 = 996). So the sum of the digits of z is 9(n - 1) + 6 and we are given that it is 186. So we have:Gmat_mission wrote:The sum of the digits of integer z is 186 and z=10^n -4. What is the value of positive integer n?
(A) 19
(B) 20
(C) 21
(D) 22
(E) 23
[spoiler]OA=C[/spoiler]
Source: Veritas Prep
9(n - 1) + 6 = 186
9(n - 1) = 180
n - 1 = 20
n = 21
Answer: C
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