Without testing numbers:

First, there are only two remainders possible when you divide n by 2: 0 and 1. The remainder is 0 if n is even, and 1 if n is odd. So the question is really just asking "is n odd?"

Remember the quotient/remainder definition. When we divide n by d, we have n = qd + r, where r is the remainder and q the quotient.

From S1, n = 5q + r, where r is odd. So n = 5q + odd, and n could be even if q is odd, and n could be odd if q is even. Insufficient.

From S2, n = 10q + r where r is odd. So n = even + odd = odd. Sufficient.
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(1) Let n = 8, remainder 3.....so, 8/2 = remainder = 0
Let n = 16, remainder 1....so 16/2 = remainder = 0
Suff
(2) n = can be 10,30,50,70,90,110,130 etc
All these numbers are divisible by 2, remainder = 0
Suff

D

P.S > I hope i'm not missing something

1) n = 5q + odd

^This doesn't tell us whether q is divisible by 2 or not. So this info is insufficient.

2) n = 10q + odd,
= 2(5q) + odd

^We can see that the remainder is odd. So this info is sufficient. B

Answer should be B.

If divided by 10 ,and the remainder is odd, then remainder will always be 1 when divided by 2
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answer is b

1. in suff
no will be like 6,8,11,13

2. suff - no will be always odd-11,13 15 , 17


so b is the answer
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Bunuel,

how can the remainder of n/5 be 11 or 6 and so on? Isn't it always between 1 and 4? e.g. 9/ 5 = 1+4 or 23 / 5 = 1+3.
What am I getting wrong here?

1. n can be 3, then the reminder will be 1, or n can be 8, and the remainder is 0. NS
2. n is odd, thus, the remainder when n is divided by 2 will always be 1.

statement 1
n/5 gives odd remainder, since 2 is not a factor of 5 we can't say about the remainder of n/2. not sufficient.

statement 2
n/10 gives odd remainder, since 2 is factor of 10(2*5),so n/2 will have same remainder which is odd.

hence B is answer.

14 divided by 5 gives the remainder of 4, not 1. Also, you are analyzing statement (1) there, which is not sufficient.

Statement (2) says: When n is divided by 10, the remainder is an odd integer
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We are not asked to find the value of n. The question asks: what is the remainder when the positive integer n is divided by 2? For all possible values of n, from (2), the reminder is 1. So, (2) is sufficient to answer the question.
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