Bunuel wrote:

If a, b, c, and z are nonnegative integers, what is the remainder when 3^(az)×3^(bz)×3^(cz) is divided by 4?

(1) z is even.

(2) The sum of a, b, and c is odd.

Hi,

3^(az)×3^(bz)×3^(cz) =\(3^{az+bz+cz} = 3^{(a+b+c)z}\)..

Two methods --

I. check for pattern -3^1 leaves a remainder 3

and 3^2 leaves a remainder 1

3^3 leaves a remainder 3

and 3^4 leaves a remainder 1..

so ODD power leaves remainder 3 and EVEN power leaves a remainder 1..

SO if we know whether POWER is ODD or EVEN, we can answer..POWER is product of a+b+c and z

lets see the statements--(1) z is even.Hence POWER is EVEN irrespective of what a+b+c is..

remainder is 1

Suff

(2) The sum of a, b, and c is odd.we do not know about z..

If z is ODD.. power is ODD and ans is 3..

But if z is Even .. power is EVEN and ans is 1

different answers possible

Insuff

ans A

Algebrically through expansion..

(1) z is even.so \(3^{(a+b+c)z} = 3^{(a+b+c)*2*y}\) as z= 2y where y is an integer, nonnegative

\((3^2)^{(a+b+c)*y} = 9^{(a+b+c)*y} = (8+1)^{(a+b+c)*y}\)..

so all terms will be div by 8 and therefore by 4 except \(1^{((a+b+c)*y}\)

so remainder = 1..

Suff

(2) The sum of a, b, and c is odd.

we cannot work further on it..

Insuff

_________________

1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372

2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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