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if M and N are integers, is (10^M + N)/3 an integer? 1. N =

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if M and N are integers, is (10^M + N)/3 an integer? 1. N = [#permalink] New post 06 Aug 2009, 19:56
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if M and N are integers, is (10^M + N)/3 an integer?

1. N = 5
2. MN is even
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Re: is it an integer? [#permalink] New post 06 Aug 2009, 21:59
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Statement 1: if u take N=5 and any value for M it will always hold true that is (10^M + N)/3 an integer; becoz 10^M ( let M = any value) will always leave a reminder of 1....and 1+5 = 6 which is divisable by 3 ....sufficient

Statement 2 : if MN is even ...this means (M,N) can be both even or odd (except both being odd at a time)....if taken an example....

(M,N) => (1,2) => 10+2 =12/3 => integer
(M,N) => (1,4) => 10+4 =14/3 => non-integer

hence insufficient

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Re: is it an integer? [#permalink] New post 06 Aug 2009, 22:13
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Let's consider 10^M

at M>=0: 10^M = {1, 10, 100, 1000....} or 3k+1
at M<0: 10^M is a fraction and our expression is not an integer.

Now, let's see our statements:

1) N=5
at M>=0: (3k+1 + 5) /3 = k+2 - an integer
at M<0: a fraction.
insufficient

2) MN is even
a) at M=2, N=5 our expression is an integer
b) at M=-2, N=-5 our expression in a fraction
insufficient

1)&2) N=5 & MN is even --> N=5, M is even
For all even M>=0 we will get an integer.
Now we have interesting question: Could negative numbers be even or odd?
if yes, we can choose M=-2, N=5 and get a fraction.
if no, M cannot be negative and two statements are sufficient.

Judging by http://en.wikipedia.org/wiki/Even_and_odd_numbers negative integers can be classified as odd/even. So, our answer is insufficient.

E
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Re: is it an integer? [#permalink] New post 06 Aug 2009, 23:14
sprtng wrote:
if M and N are integers, is (10^M + N)/3 an integer?

1. N = 5
2. MN is even


Walker got it , great job
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Re: is it an integer? [#permalink] New post 07 Aug 2009, 05:33
yup. that's the answer.
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Re: is it an integer? [#permalink] New post 17 Dec 2009, 19:25
how did you get 3k + 1? can someone please explain? thanks.
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Re: is it an integer? [#permalink] New post 18 Dec 2009, 20:18
10^K = 3k+1: for any K>=0
It means any number which is 10^K i.e. 1,10,100,1000 ..., when divide by 3, gives remainder 1.
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Re: is it an integer? [#permalink] New post 18 Dec 2009, 20:30
thanks, obviously i still have a lot of catching up to do when it comes to math fundamentals.
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Re: is it an integer? [#permalink] New post 22 Jan 2010, 23:24
yeah, i made the same mistake of opting for A.

but it's E, because we don't no whether m and n are +ve integers
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Re: is it an integer? [#permalink] New post 23 Jan 2010, 09:56
Thanks I committed 2 careless mistakes

Initially I chose A ignoring the fact M and are just integers not positive integers

and then I chose C since MN is even I just assumed it is positive.

but the correct ans is E since both statements would be insufficient to ans the question
Re: is it an integer?   [#permalink] 23 Jan 2010, 09:56
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