If 3 is a factor of positive integers x and y, is x odd?

Question

If 3 is a factor of positive integers  x  and  y , is  x  odd?

(1) Units digit of  4^{x+y}  is 6

(2) Units digit of  9^{xy}  is 9

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divyadna · Answer

Is x odd?
x=3a and y=3b

(1) Units digit of 4^x+y is 6. this means that x+y is even. 
even+even=even e.g. 6+6=12. x is even here. Answer is NO
OR
odd+odd= even e.g. 3+9=12. x is odd here. Answer is YES.
Statement I ALONE IS NOT SUFFICIENT.

(2) Units digit of 9^xy is 9
xy= odd
odd*odd=odd
so, x is definitrly odd. 
Statement II alone is sufficient.
Option B is the answer.

$!vakumar.m · Answer

If 3 is a factor of positive integers x and y, is x odd?
x and y are multiples of 3... which can be either odd or even

(1) Units digit of 4^(x+y) is 6
4 has cyclicity of 2, for odd powers of 4, the units digit is 4 and for even powers of 4, the units digit is 6
hence, x+y is Even
=> both x and y are of same sign (i.e., both even or both odd)
Not sufficient
(2) Units digit of 9^(xy) is 9
9 has cyclicity of 2, for odd powers of 9, the units digit is 9 and for even powers of 9, the units digit is 1
hence, xy is Odd
=> both x and y are Odd
Sufficient
Answer B

walterwhite756 · Answer

Given 3 is a factor of positive integers  x  and  y , so  x  and  y  are multiples of 3.

Stmt 1: Units digit of  4^{x+y}  is 6.
Focus on units digit when 4 is raised to nth power and  keep multiplying units digits with base number.  
4^1 = 4, 4^2 = 6, 4^3 = 4, 4^4 = 6 and so on. Basically 4 when raised to some power, it will have unit's digit =  4  if power is  odd  and =  6  if power is  even .

So x + y = even.

Now, either both x and y are even or both x and y are odd. Nothing can be said further hence  insufficient .

Stmt 2: Units digit of  9^{xy}  is 9

Focus on units digit when 9 is raised to nth power and  keep multiplying units digits with base number. 
9^1 = 9, 9^2 = 1, 9^3 = 9, 9^4 = 1 and so on. Basically 9 when raised to some power, it will have unit's digit =  9  if power is  odd  and =  1  if power is  even .

Therefore xy has to be odd. Only case when this is possible is, both x and y are odd. Hence  sufficient .

Therefore answer is B.

If 3 is a factor of positive integers x and y, is x odd?

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GMAT Data Sufficiency (DS) Questions

If 3 is a factor of positive integers x and y, is x odd?

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