An increasing sequence M consists of 5 consecutive positive multiples

Since the sequence consists of consecutive positive multiples, we can consider the following cases:
Case 1: {even, odd, even, odd, even} or
Case 2: {odd, even, odd, even, odd} or
Case 3: {even, even, even, even, even}

Therefore, we know that the greatest integer can be odd or even and hence the remainder can be 1 or 0.

From statement I alone, we know that the median of the sequence is even. This means that the set can correspond to either case 1 or case 3. In both cases, the largest integer is even and hence the remainder is 0.
Statement I alone is sufficient to answer the question. Answer options B, C and E can be eliminated. Possible answer options are A or D.

From statement II alone, we know that the second term of the sequence is odd. This means that the set corresponds to case 1 and hence the largest integer is even. The remainder when this integer is divided by 2 will be 0.
Statement II alone is sufficient. Answer option A can be eliminated.

The correct answer option is D.

Hope that helps!

Set M can be ( 2,4,6,8,10) ; ( 3,6,9,12,15) ( 6,9,12,15,18) ( 10,15,20,25,30)

#1The median of the sequence is even.

( 2,4,6,8,10) ; ( 6,9,12,15,18)
largest term will be even so remainder is 0 when divided by 2; sufficient
#2

The second term of the sequence is odd.
( 6,9,12,15,18) ;( 10,15,20,25,30)
largest term will be even so remainder is 0 when divided by 2; sufficient
OPTION D

5 consecutive multiples of a positive integer
Multiples follow the order -> odd, even, odd, even, odd......

Statement 1 => Median is even, means 5th term is even. Hence the remainder is 0 => Sufficient
Statement 2 => 2nd number is odd, means 5th term is even. Hence the remainder is 0 => Sufficient.

Answer is D.

ST1: Let the median be defined as X*Y. Since X*Y is even, so at-least one of them is even. If X is even, then all the terms of the sequence are even. If Y is even, then the largest term X*(Y+2) will also be even. Thus, r=0 always.

ST2: 2nd term is odd. Let 2nd term X*Y. This means both X and Y are odd. Y is odd means Y+3 will be even. So largest term given by X*(Y+3) will be even. Thus, r=0.

Ans: D

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