Numbers

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shankar.ashwin: Legendary Member; Posts: 966; Joined: Sat Jan 02, 2010 8:06 amFollowed by: 21 members; Thanked: 230 times

by shankar.ashwin » Wed Sep 21, 2011 9:27 am

Let S(n) denote the sum of the digits of a natural number 'n'.
2n + S(n) = 1998. Then S(n) can be :

A) 24
B) 21
C) 17
D) 22
E) 20

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Source: Beat The GMAT — Problem Solving | Wed Sep 21, 2011 9:27 am

cans: Legendary Member; Posts: 1309; Joined: Mon Apr 04, 2011 5:34 amFollowed by: 123 members; Location: India; Thanked: 310 times; GMAT Score:750

by cans » Wed Sep 21, 2011 9:45 am

2n + S(n) = 1998.
C) 17 not possible. then sum will be odd.
E) 20. n=989. S(n)!=20
D) 22. n=988 not
A) 24. 987. sum=24
IMO A

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gmatclubmember: Master | Next Rank: 500 Posts; Posts: 158; Joined: Sat Sep 03, 2011 10:31 amFollowed by: 2 members; Thanked: 29 times

by gmatclubmember » Wed Sep 21, 2011 10:25 am

The detailed solution to this question would be like this:
lets say the no. is xyz (it cant be a 2 digit no. otherwise the summation wont add upto 1998).
as per the equation 2n+S(n)=1998=> 2(100x+10y+z) + x+y+z=1998
=>201x+21y+3z=1998
=>67x+7y+z=666
To solve above equation:
Lets assume the max value of y and z (which is 9) and it would give us 67x=594
and thus we get the least integral value of x=9.
7y+z=63.
the solution set for (y,z) would be (9,0) and (8,7).
First set gives us the value of n as 990 which doesnt satisfy the eqn above.
Second set gives us the value 987 which satisfy the eqn above.
S(n) is 24.

Cheers
Ami/-

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tpr-becky: GMAT Instructor; Posts: 509; Joined: Wed Apr 21, 2010 1:08 pmFollowed by: 85 members; Location: Irvine, CA; Thanked: 199 times; GMAT Score:750

by tpr-becky » Wed Sep 21, 2011 3:10 pm

This is one in which I would look at the answers to help if 2n + S(n) = 1998 that means that
1998 - S(n) = 2n, which means that 1998 - S(n) will be even. Then we can eliminate the odd . answers becuase if we subtract an odd number from 1998 we will get an odd number.

It is acutally pretty easy to look at the answers:

24) 1998 - 24 = 1974 - divide that by 2 to get 987 add the digits of 987 and you get 24 - thus this answer works.

22) 1998 - 22 = 1978 - divide by 2 to get 988 - the sum of the digits of 988 is 25 (this doesn't work)

20) 1998 - 20 = 1978 divide by 2 to get 989 - sum of digits is 26 - doesn't match.

Becky
Master GMAT Instructor
The Princeton Review
Irvine, CA

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