If a, b, c and d are integers, how many of these are odd?

Question

If a, b, c and d are positive integers, how many of these are odd?

(1) a+b=c+d+2
(2) a+b < 6

ArunSharma12 · Answer

a,b,c & d > 0

statement 1:
a+b=c+d+2
this tells us number of odd numbers should be even.
either 0,2 or 4 number of given numbers (a,b,c,d) can be odd.
not sufficient

statement 2:
a+b < 6
possible pairs of (a,b): (1,2),(1,3),(1,4),(2,2)(2,3)
but we do not know values of c & d.
not sufficient

combining both statements,
 (1,2): c+d =1; not possible because numbers are positive
(1,3): c+d=2; (c,d): (1,1)-> number of odd numbers: 4
(1,4): c+d=3; (c,d): (1,2)-> number of odd numbers: 2
(2,2): c+d=2; (c,d): (1,1)-> number of odd numbers: 2
(2,3): c+d=3; (c,d):(1,2)-> number of odd numbers: 2
 
Ans: E

unraveled · Answer

If a, b, c and d are distinct positive integers, how many of these are odd?
a,b,c and d > 0

(1) a+b=c+d+2
CASE I: o + o = o + o + 2 = e ---> n(odd) = 4
CASE II: o + o = e + e + 2 = e ---> n(odd) = 2(since we don't know what values each of them hold.)

INSUFFICIENT.

(2) a+b < 6
CASE I: a + b = 2 + 3 = 5 ----> n(odd) = 4 if c and d are also odd.
CASE II: a + b = 2 + 3 = 5 ----> n(odd) = 2 if c and d are even.

INSUFFICIENT.

Together 1 and 2
Since a + b < 6,
c + d < 4
Thus, 1 + 2 < 4 OR 2 + 1 < 4 in either case only one is odd
So,
a + b = 1 + 2 + 2 = 5
a and b can't take 1 and 2 as their values since the are distinct positive integers.
But either of a and b also can't take values greater than 4.
Hence Possible values are 3 and 4 only. However, this invalidates the condition as:
a + b = 3 + 4 > 4

Answer E.

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GMATinsight · Answer

Question : How many of a, b, c, and d are ODD? Statement 1 : a+b=c+d+2 Adding 2 in any integer does NOT change the property of a number in terms of result being even or ODD i.e. The expression can also be considered as a+b=c+d Case 1: a+b=c+d+2 = Odd then one of a and b must be even and other must be odd and one of c and d must be even and other must be odd i.e. 2 even and 2 odd integers Case 2: a+b=c+d+2 = Even then both a and b must be even or both must be odd and both c and d must be even or both must be odd i.e. 2 even and 2 odd integers OR All 4 EVEN integers or All 4 ODD integers or NOT SUFFICIENT Statement 2 : a+b < 6 No information about c and d hence NOT SUFFICIENT Combining the statement s We know that a, b, c and d are distinct positive integers and also, a+b < 6 and a+b=c+d+2 SInce c and d minimum can be 1 and 2 (positive distinct integers) therefore, a+b minimum should be 5 i.e. a+b = 5 (because it's also less than 6) i.e. one of a and b is even and other is odd (2,3) or (1,4) now, c+d = a+b-2 = 5-2 = 3 i.e. c and d must be (1, 2) in any order i.e. ONE EVEN and ONE ODD now a and b must have one even and one odd integer too so we will have a total 2 ODD integers Answer: Option C chetan2u With the given inputs, a, b, c and d can NOT be positive distinct integers. Please check

ShankSouljaBoi · Answer

Great question !!! However , i am not able to make a pair of numbers that will satisfy a, b , c and d to be distinct integers.

I'll still go for C on this one. Waiting for official explanation...

chetan2u · Answer

Sorry, typo from my side. 
Edited. ?

unraveled · Answer

Guess that highlighted part showed something was not right with the question. So, one learning is that even if, if ever, something is not right in such type of question, the answer would not have changed. I had two odd integers..aargh..!!

freedom128 · Answer

a, b, c and d are positive integers

Q. How many of these are odd?

(1) a+b=c+d+2

- If (c+d) is odd, then (a+b) is odd.
 Two  of a, b, c, and d are odd, i.e. 
c+d=1+2=3 and a+b=2+3=5

- If (c+d) is even, then (a+b) is even. 
 None or two or all  of a, b, c, and d are odd, i.e. 
c+d=1+1=2 and a+b=1+3=4 (4 odd)
c+d=1+1=2 and a+b=2+2=4 (2 odd)
c+d=2+2=4 and a+b=3+3=6 (2 odd)
c+d=2+2=4 and a+b=2+4=6 (0 odd)

We cannot deduce how many of these are odd. 
NOT SUFFICIENT

(2) a+b < 6 
(a+b) can be from 2 to 5. Thus, there must be some variation of how many of these (a, b, c, d) are odd.
NOT SUFFICIENT

Combined 
a+b = c+d+2 < 6 --> (c+d)<4
(c+d) can be 2 or 3.

If (c+d)=1+2=3, then (a+b)= 5 = 1+4 or 2+3. 
 Two  of a, b, c, and d are odd.

If (c+d)=1+1=2, then (a+b)= 4 = 2+2 or 1+3. 
 Two or four  of a, b, c, and d are odd.

We still cannot deduce how many of these are odd. 
NOT SUFFICIENT

FINAL ANSWER IS (E)

chetan2u , I think the correct answer is (E). Please clarify

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If a, b, c and d are integers, how many of these are odd?

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