If n is a positive integer, is 12 a factor of n×(n+1)×(n+2)

Question

If  n  is a positive integer, is  12  a factor of  n×(n+1)×(n+2) ?
1)  n×(n+2)  is odd
2)  n+1  is even

CareerGeek · Answer

n×(n+1)×(n+2) 
--> 2 possibilities: 
1st case: odd*even*odd --> {1*2*3 or 3*4*5 or 5*6*7, 11*12*13} = divisible by 6 as well as 12
2nd case: even*odd*even --> {2*3*4 or 4*5*6 or 6*7*8} = ALWAYS divisible by 24

1) n×(n+2) is odd 
--> n is odd
--> 1st case -  Insufficient

2) (n+1) is even
--> n is odd
--> 1st case -  Insufficient

Combining (1) & (2)
Gives same case that n is odd

-  Insufficient

IMO Option E

Pls Hit Kudos if you like the solution

nick1816 · Answer

IMO E is the answer!!!

Statement 1
If n*(n+2) is an odd number, then both n and n+2 are odd which implies n+1 is an even number
If n+1 is an even multiple of 2,  12  is a factor of  n×(n+1)×(n+2)

But if n+1 is an odd multiple of 2, then  12  is not a factor of  n×(n+1)×(n+2)

Insufficient

Statement 2
If n+1 is even, both n and n+2 are odd numbers
If n+1 is an even multiple of 2,  12  is a factor of  n×(n+1)×(n+2)

But if n+1 is an odd multiple of 2, then  12  is not a factor of  n×(n+1)×(n+2)

Combining both equations 
Both statements implying that n+1 is even, but none tells us that n+1 is even multiple of 2
Insufficient!!

CareerGeek · Answer

IMO E is the answer!!!

Statement 1
If n*(n+2) is an odd number, then both n and n+2 are odd which implies n+1 is an even number
If n+1 is an even multiple of 2,  12  is a factor of  n×(n+1)×(n+2)

But if n+1 is an odd multiple of 2, then  12  is not a factor of  n×(n+1)×(n+2)

Insufficient

Statement 2
If n+1 is even, both n and n+2 are odd numbers
If n+1 is an even multiple of 2,  12  is a factor of  n×(n+1)×(n+2)

But if n+1 is an odd multiple of 2, then  12  is not a factor of  n×(n+1)×(n+2)

Combining both equations 
Both statements implying that n+1 is even, but none tells us that n+1 is even multiple of 2
Insufficient!!

Yeah I have left a case in (1) & (2) which is 11*12*13
Thanks for the correction
Kudos!

BelalHossain046 · Answer

Statement 1: 
Case 1: n=1 & (n+2)=3.   So, (n+1)=2. Here, n×(n+1)×(n+2)=1*2*3=6  NOT DIVISIBLE BY 12 ]
Case 2: n=3 & (n+2)=5.   So, (n+1)=4. Here, n×(n+1)×(n+2)=3*4*5=60   DIVISIBLE BY 12 ]
NOT SUFFICIENT.

Statement 2: SAME TWO CASES of Statement 1 are applicable here.
Case 1: n+1=2 So, n=1 & (n+2)=3. Here, n×(n+1)×(n+2)=1*2*3=6  NOT DIVISIBLE BY 12 ]
Case 2: n+1=4 So, n=3 & (n+2)=5. Here, n×(n+1)×(n+2)=3*4*5=60   DIVISIBLE BY 12 ]
NOT SUFFICIENT.

COMBINING: AS SAME NUMBER MADE BOTH STATEMENTS INSUFFICIENT, COMBINING is also INSUFFICIENT.
Both   &   suffice both statements. 
 Case 1:n=1, n+1=2 & (n+2)=3. Here, n×(n+1)×(n+2)=1*2*3=6  NOT DIVISIBLE BY 12 ]
Case 2: n=3, n+1=4 & (n+2)=5. Here, n×(n+1)×(n+2)=3*4*5=60   DIVISIBLE BY 12 ]
INSUFFICIENT.
E

VISHNUNAMBI · Answer

by combining statement 1 and statement 2 we get that the product is always odd.

And we know clearly that 12 cannot be a factor of any odd integer , clearly we can say 12 is not the factor of given number

so it correct answer should be C

bumpbot · Answer

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If n is a positive integer, is 12 a factor of n×(n+1)×(n+2)

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If n is a positive integer, is 12 a factor of n×(n+1)×(n+2)

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