M is a positive integer, is M odd?

Question

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

14101992 · Answer

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 and if M is off (M^2+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

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

Statement 1 Sufficient.

Statement 2: M+10 mod 10 = 0
So, M has to be a multiple of 10 for above to be true and all multiple of 10 are even.

Statement 2 satisfies.

Answer D.

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

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Nightfury14 · Answer

Statement 1) 
Case 1 :: M=1 => 2x 1^3 +2x1 = 4 .. Not divisible by 8 
Case 2 :: M=4 => 2x 4^3 +2x4 = 136 .. Divisible by 8 .. Not sufficient

Statement 2) M is multiple of 10 .. M is even .. Sufficient

Anser Choice B)

Good luck
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14101992 · Answer

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.

Nightfury14 · Answer

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..

ashwini86 · Answer

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

pallaviisinha · Answer

Hi,

Can someone explain in detail why statement 1 is sufficient.

Thanks.

14101992 · Answer

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

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.

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M is a positive integer, is M odd?

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M is a positive integer, is M odd?

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