If a is divisible by 2, is b + 5 an integer?

Question

(1) The median of a and b is not an integer.
(2) The average (arithmetic mean) of 3a, b, and b + 10 is an even integer.

Something that occasionally throws me off and I often forget. When we get a variable on the gmat, what can we assume about it? Just that it is a real, rational number? We CANNOT automatically assume it's an integer correct?

NonYankee · Answer

Right. We only know that b is real  and rational . If we could assume variables were integers, we wouldn't need any additional statements at all to answer the question, "Is b + 5 an integer?"

jbisht · Answer

answer should be Option(b) as

from (1) :- either b is an odd integer or a real number of type x.y ...........hence not sufficient.
from (2) : (3a+b+b+10)/3 = 2k (let say)
 so, 3a + 2b = 6k - 10
 b = (3(2k-a) - 10)/2
b= 3(2k - a)/2 -5
as k is even, so 2k is also even also even - even is an even hence (2k-a) is divisible by 2
so, b = 3*some_integer - 5
hence b = integer...............hence condition 2 is sufficient

hence answer should be option(b)

NonYankee · Answer

I bolded two parts of your explanation. You originally indicated that 2k was even. k itself may be even or odd. You later asserted that k was even.

mau5 · Answer

I don't think the given OA is correct. Anyways, let a = 2p,where p is any integer.

From F.S 1, we have that \frac{(a+b)}{2}  = Not an integer

or \frac{2p}{2}+\frac{b}{2} = p+\frac{b}{2} = Not an integer. Assuming, b = 3 , we satisfy the fact statement and get a YES for the question stem. Again assuming b=1.2, we satisfy the fact statement but get a NO for the question stem.Insufficient.

From F.S 2, we have  3a+b+(b+10)/3  = 2k, where k is again an integer.

Thus, 3a/3 + (b+5)/3+(b+5)/3  = 2k

or 2/3*(b+5) = 2k-a = 2k-2p = 2(k-p)

or (b+5) = An integer*3 = integer.

Sufficient.

B.

Bunuel · Answer

From the stem we cannot assume that b is a rational number, it could be an irrational as well, for example  \sqrt{2} .

On the GMAT, if no other constraint is given about a variable, then all we can say that it's a real number.

If a is divisible by 2, is b + 5 an integer?

Notice that:
a is divisible by 2 implies that a is an even number.
The question basically asks whether b is an integer.

(1) The median of a and b is not an integer. b may or may not be an integer. Consider: a=2 and b=3 for an YES answer and a=2 and b=1/2 for a NO answer. Not sufficient.

(2) The average (arithmetic mean) of 3a, b, and b + 10 is an even integer -->  3a+b+(b+10)=3*even  -->  2b+10=3*even-3a  --> since a is even, then  2b+10=3*even-3*even  -->  2(b+5)=3(even-even)=3*even  -->  b+5=3*\frac{even}{2}=3*integer=integer . Sufficient.

Answer: B.

Hope it's clear.

NonYankee · Answer

You make a good point. In any case, it's not an integer.

Divyadisha · Answer

a = even

b+5 is an integer?

(1) The median of a and b is not an integer.

a+b/2 is not an integer.

Its given that a is even. The above expression can not be an integer if b is odd or fraction.

(2) The average (arithmetic mean) of 3a, b, and b + 10 is an even integer

3a+b+b+10/2= integer

3a+2b+10/2= integer

3a is even (a is even), 2 b is even and 10 is even. To have above equation as an integer b must be an integer.

Hence b+5 will be an integer.

B is the answer

niks18 · Answer

From the question stem it is clear that  a  is  Even  and we need to know whether  b  is integer

Statement 1:  implies  \frac{a+b}{2} =  Non Integer (NI). if  a=2  &  b=1.5 , then median is non integer and  b+5  is also a non integer

but if  a=2  &  b=3 , then median is non integer but  b+5  is an integer. Hence we have both a Yes & a NO.  Insufficient

Statement 2:  implies  \frac{3a+b+b+10}{3}=Even => 3a+2b+10=3*Even

=>2b=Even-10-3a . Now as  a  is even so  3a  is even and  10  is also even

so  2b=Even-Even-Even=Even => b=\frac{Even}{2}=Integer . Hence  b+5  will be an integer.  Sufficient

Option  B

milanrajb · Answer

I have a major confusion here.

Everyone here is assuming that a is an int. A number divisible by 2 can be a non-integer as well. Ex - 2.2

Bunuel · Answer

Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers (ALL GMAT divisibility questions are limited to positive integers only). So, you are right, a proper GMAT question should have explicitly mentioned that  a  is a positive integer. But as far as GMAT is concerned,  a  is divisible by  b  means that:

(i)  a  is an integer;
(ii)  b  is an integer;
(iii)  \frac{a}{b}=integer .

So, 2.2 is divisible by 2 does not make sense because 2.2 is NOT an integer.

Hope it helps.

If a is divisible by 2, is b + 5 an integer?

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If a is divisible by 2, is b + 5 an integer?

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