If m and n are positive integers greater than 5 and (m^2 + 1)(n^2 +1)

Question

If m and n are positive integers greater than 5 and (m^2 + 1)(n^2 + 1) is odd, what is the greatest positive integer that always divides mn? A. 1 B. 2 C. 4 D. 8 E. 16 2024-01-28_15-45-16.png

chetan2u · Accepted Answer

(m^2 +1)(n^2 +1)   is odd  
=> Both  (m^2 +1)  and  (n^2 +1)  are odd, since only possibility for a product to be odd is Odd*Odd=Odd
Thus, m and n are even, => m=2x and n=2y. Thus mn = 4x, or a multiple of 4 for sure.

C

dpchen · Answer

For those who got m=6 and n=8 and thought the answer should be E (including myself...), I think we cannot rule out m = n = 6, which will make mn = 36, which is not divisible by 8 or 16.

shivanks · Answer

(M^2+1)(N^2+1) is odd which means m and n both are even.
or according to question m and n is greater than 5 which means, m is possibly 6 and n is possibly 8,
let,s try (6^2+1)(8^2+1)= (37)(65) product of them is must odd and in that situation,
mn= 6*8= 48 multiple of 2. 
in my way the ans is 2 option b

sachin_27 · Answer

taking m and n 6 and 8. 
(m^2+1)(n^2+1) = 37*65 = odd (condition satisfied)
now m*n = 48, so according to options, the largest possible integer to divide 48, should be 16. why is it 4? 
please someone explain?

wildbacon07 · Answer

The answer is c) 4:

Why? (m^2+1)(n^2+1) is odd which means that both (m^2+1) and (n^2+1) must be odd individually.

In order to comply with this, both m and n need to be even. If both are even and >5 then 2 will be a factor of both of them so they can be written as m=2x and n=2y.

If we multiply them m*n=2x*2y=4xy which means that the resultant number will always be divisible by 4 (think of it as a consequence of both m and n having a factor 2 in their prime factorization).

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Oppenheimer1945 · Answer

(m^2 + 1)(n^2 + 1)  is odd means both  m   &  n   are even.  mn  is always divisible by greatest possible  X , Least possible.

mn  is  2a ,  2b  respectively. product is always div by  2*2=4  when   gcd(a,b)=1

siddd30 · Answer

We can clearly see that M and N are even, since the product of odd numbers can only yield an odd answer
We also know that, both M and N are greater than 5. Taking values for M and N starting with the least ones:

M, N= 6

M*N= 36, Highest factor available from the options is 4

Since, its an "always divides" question- then this pretty much answers

RM4321 · Answer

Why it can't be 8 . I know it can be 4. I want to understand the reason behind not 8 ? Is it that the numbers can be same ?

Bunuel · Answer

Yes, m and n can be the same number. For instance, if both are 6, then mn = 36, which is not divisible by 8. But that’s not the main point. If both m and n are even but not multiples of 4, then mn will still not be divisible by 8. For example, m = 14 and n = 22.

If m and n are positive integers greater than 5 and (m^2 + 1)(n^2 +1)

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If m and n are positive integers greater than 5 and (m^2 + 1)(n^2 +1)

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