If M and N are positive integers greater than 1, what is the remainder

Question

If M and N are positive integers greater than 1, what is the remainder when the expression  22^{3M} × 39^{2N} + 14^{2(M+N)}  is divided by 5?
(1) M = 13
(2) N = 14

GMATinsight · Answer

A beautiful Question Please find below the video explanation Answer: Option A https://www.youtube.com/watch?v=qTQkdT1y5Gs My YouTube Video Treasure https://chat.whatsapp.com/EosbbxhyMjRLtqNGMUm8Yy

Kinshook · Answer

223M × 392N + 142(M+N)mod5= (6MN+2M+2N)mod5; 2(3MN+M+N)mod5;

(1)M=13; 
2(39N+13+N)mod5 = (80N+26)mod5 = 1; 
SUFFICIENT;

(2) N=14; 
2(42M+14+M)mod5 = (86M + 28)mod5 = (M+3)mod5; 
NOT SUFFICIENT;

IMO A

Kritisood · Answer

Hi, I had mistakenly not put the powers correctly  GMATinsight   Kinshook 
Have made the edit

Bunuel · Answer

If M and N are positive integers greater than 1, what is the remainder when the expression 22^3M × 39^2N + 14^2(M+N) is divided by 5?

Are the powers there 3M, 2N and 2(M+N)? If yes, then ti should be written as 22^(3M) × 39^(2N) + 14^(2(M+N)) or using formatting as  22^{3M} × 39^{2N} + 14^{2(M+N)}

Kritisood · Answer

Done.. apologies for the confusion!!

Archit3110 · Answer

using cyclicity rule we can solve this question 
for given expression  22^{3M} × 39^{2N} + 14^{2(M+N)} 
we can note that the unit digit value of 39^{2N} would always be 1 ; because its 9^ even power and unit digit value of 14^{2(M+N) would always be 6 as its also raised to even power
so the remainder value mainly depends what is the value of M for  22^{3M}  ; as M>1 ; we can get 4,1,6,8 as digits for M= 2,3,4,5 
#1
M=13
 unit digit value of  22^{3*13}  ; 8
so for given expression we get 22^(3M) × 39^(2N)+ 14^(2(M+N)
22^39 * 1 + 6
8*1+6 ; 14 ; remainder will be 4 when divided by 5
sufficient
#2
N=14 
as stated in above ; knowing value of N is not required as it would always be even power raised or say 6
so insufficient as M is not know
IMO A is correct

GMATinsight · Answer

Kritisood Without edit too the answer to the question was Option A and the question was good. I guess I will make post of that question which I solved.

bumpbot · Answer

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If M and N are positive integers greater than 1, what is the remainder

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If M and N are positive integers greater than 1, what is the remainder

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