M is a positive integer, is M odd?

Nightfury14
In statement 1, no conclusion can be drawn using case 1 and case 2. To make statement 1 insufficient, a case has to be shown in which M=odd satisfies the condition 2M^3 + 2M is divisible by 8. Which is not possible.

And hence the answer would be D.

Yeah .. you are right
I solved the question on timer, then while writing the answer lost track of what was the criteria.
Thanks for pointing out..

M is a positive integer, is M odd?

(1) 2M^3 + 2M is divisible by 8.
(2) M + 10 is divisible by 10.


A : we can simplify it to : M(M^2 + 1) is divisible by 4 , this does not hold for odd values of M hence its sufficient

B this says that M is a multiple of 10 , which means M is even

My Choice : D , lets wait for the OA

Hi,

Can someone explain in detail why statement 1 is sufficient.

Thanks.

pallaviisinha

Statement 1: 2M^3 + 2M mod 8 = 0 -----> 2M(M^2+1) mod 8 =0 -----> M(M^2+1) mod 4 = 0

We know that square of even is even and square of odd is odd. So, if M is even, (M^2+1) will be odd (as even*even+1 is odd)
and if M is odd (M^2+1) will be even (as odd*odd +1 will be even.

For M(M^2+1) mod 4 = 0 to be true we want one of M or (M^2+1) to be divisible by 4.

For even M, such as M=4, we can have M(M^2+1) mod 4 = 0

For any odd M... (M^2+1) mod 2 will be = 0 but (M^2+1) mod 4 will never be = 0

For ex. M=3 ---> M^2+1=10, M=5 ---> M^2+1=26, M=7 ---> M^2+1=50, M=9 ----> M^2+1=82
[all (M^2+1) mod 4 is not equal to 0]

Hence, we can conclude that M has to be even in order to have M(M^2+1) mod 4 = 0

Therefore, Statement 1 is sufficient.

------------------------------------

P.S. Don't forget to give kudos if you got the explanation.

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.