If m and n are positive integers, is (mn+1+3)(mn+2+4)(mn+3+5) divisibl

Question

If m and n are positive integers, is  (m^{n+1}+3)(m^{n+2}+4)(m^{n+3}+5)  divisible by 12?

1) m is an odd number
2) n is an even number

askul2389 · Answer

* *

We need to find whether the equation is divisible by 2*2*3. So the equation should have atleast 2 multiples of 2 and 1 multiple of 3.
Option A :
m is odd. Hence ; 
first term =   = even (multiple of 2)
third term =   = even (multiple of 2)

Now in second term lets assume "n" is odd and m =odd (1,3,5)

So for m =1 ;   =   = 5 ..... hence the equation is not divisible by 3.

when m = 3 ; n = 1.........   = 5 .......hence the equation is not divisible by 3.

when m = 5 ; n = 1.........   = 129 /3 = 43.......hence the equation is DIVISIBLE by 3.

So how can we conclude the equation is divisible by 12 when m is odd??

I think answer should be C.

sleepynut · Answer

1) m is an odd number
 m^{n+1}  +3 is even,hence it is divisible by 2
 m^{n+3}  + 5 is even,hence it is divisible by 2

1.1)  note the remainder of (x+y)/3 = the remainder of x/3 + the remainder of y/3  
if the remainder of m/3 is 1,then  m^{n+3} +5 is divisible by 3
if the remainder of m/3 is 2,then  m^{n+1} +4 is divisible by 3
if the remainder of m/3 is 0,then  m^{n+1} +3 is divisible by 3
Thus,(1) is SUFFICIENT

2)n is an even number
if m is even, m^{n+2} +4 is divisible by 4
if m is odd, m^{n+1} +3 and  m^{n+3} +5 is even

From (1.1),this expression always divisible by 3,(2) is also SUFFICIENT
ans (D)

gvvsnraju@1 · Answer

I request expert's reply on my approach.

While solving the question for 2nd Statement:
 Given : n is even.
m can be either odd or even.

For all values of m = odd, n = even , i have arrived that given statement is sufficient to answer.
But
For all values of m = even, n = even.

say m=n=2, Given expression =  * *  = 11*20*37 = not divisible by 12.

So, i thought the answer was A.

But the OA is D.
Let me know where i am faltering.

Thanks in advance.

Raju

shekyonline · Answer

m=n=2, Given expression =  * *  = 11*20*37 = not divisible by 12. 
  i also agree that ans. will be A.

If m and n are positive integers, is (mn+1+3)(mn+2+4)(mn+3+5) divisibl

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