is (10^M + N)/3 an integer?

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sanju09: GMAT Instructor; Posts: 3650; Joined: Wed Jan 21, 2009 4:27 amFollowed by: 80 members; Location: India; Thanked: 267 times; GMAT Score:760

by sanju09 » Sat Aug 14, 2010 4:50 am

If M and N are integers, is (10^M + N)/3 an integer?

1. N = 5.
2. MN is even.

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Source: Beat The GMAT — Data Sufficiency | Sat Aug 14, 2010 4:50 am

kvcpk: Legendary Member; Posts: 1893; Joined: Sun May 30, 2010 11:48 pmFollowed by: 7 members; Thanked: 215 times

by kvcpk » Sat Aug 14, 2010 5:18 am

sanju09 wrote:If M and N are integers, is (10^M + N)/3 an integer?

1. N = 5.
2. MN is even.

[spoiler]Source: Picked from some source unknown to www.avenuesabroad.org[/spoiler]

(10^M + N)/3 is integer??

1. N = 5.
10^M + 5
when m=-1, 5.1/3 is not integer.
when m=1, 15/3 is integer.
INSUFF

2. MN is even.
when m=0, n=5, 6/3 is integer.
when m=-1, n=2, 2.1/3 is not an integer.
INSUFF

Combining:
MN is even and n=5
Hence M is even.
when m=0, n=5, 6/3 is integer.
when m=-2,n=5, 5.01/3 is not an integer.
INSUFF

pick E.

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reply2spg: Legendary Member; Posts: 1261; Joined: Sun Sep 14, 2008 3:46 am; Thanked: 27 times; GMAT Score:570

by reply2spg » Sat Aug 14, 2010 5:19 am

Only A is correct here

Sudhanshu
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chris@veritasprep: GMAT Instructor; Posts: 31; Joined: Tue Jul 27, 2010 7:09 amFollowed by: 25 members; Thanked: 49 times

by chris@veritasprep » Sat Aug 14, 2010 7:00 am

If the OA is A then this is clearly a flawed question (but a good one when the flaw is removed!) - the question stem would have to tell you that M and N are non-negative integers as statement one is not sufficient if, for instance, M = -1 or -2. 1/10 + 5 is not divisible by 3 as kvpck pointed out!

If m is an integer >= 0 then statement 1 is surely sufficient and the answer would be A. As it is with the wording now, the answer would indeed be E. Lets do the problem if M and N are non-negative integers as I suspect that was the intention of the problem:

Anytime that you divide a non-fractional power of 10 by three the remainer will be 1. Therefore if you add 5 to any power of 10 it will always be divisible by 3.

An even clearer way to think about this problem is with an understanding of the divisiblity rules. If the sum of the digits of a number add up to a mulitple of 3, then the whole number is divisible by 3. The sum of the digits for all non-fractional powers of 10 would be 1. By adding 5 to that then you know the sum of the digits of all non-fractional powers of 10 +5 would be 6 and therefore always divisible by 3.

If confused on these two concepts then number picking would also be quite effective (a strategy that I don't much care for but it has its place!)

Make M = 0 so 10^0 = 1 + 5 = 6 divisible by 3
Make M = 1 so 10^1 = 10 +5 = 15 divisible by 3
Make M = 2 so 10^2 = 100 + 5 = 105 divisible by 3

At this point the number picking should help clarify the two concepts discussed above and the answer would be A. With almost all GMAT problems you want to first try to solve them with an understanding of concepts or by using algebra. If that is failing you then move on to number picking and see if you can establish a pattern and find the answer. NOTE: some problems (hard quotient/remainder for instance) are sometimes best solved first with number picking. Hope this helps!

Chris Kane
GMAT Instructor
Veritas Prep

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