An increasing sequence M consists of 5 consecutive positive multiples

Question

An increasing sequence M consists of 5 consecutive positive multiples of a positive integer. What is the remainder when the largest term of the sequence is divided by 2?

(1) The median of the sequence is even.

(2) The second term of the sequence is odd.

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firas92 · Answer

Let the sequence be { an, a(n+1), a(n+2), a(n+3), a(n+4) } where  a  and  n  are positive integers.

(1) The median of the sequence is even.

a(n+2)  is even. So either  a  is even or  (n+2)  is even or both  a  and  (n+2)  are even.

In any of these cases  a(n+4)  must be even and so leaves a remainder 0 when divided by 2.

1 is sufficient

(2) The second term of the sequence is odd.

a(n+1)  is odd. So both  a  and  (n+1)  are odd. This means  (n+4)  must be even which makes  a(n+4)  even and so leaves a remainder 0 when divided by 2

2 is sufficient

Answer is  (D)

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GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • Let us assume that the positive integer is n and
 o Sequence M  = (m- 2)*n , (m-1)*n, m*n, (m+1)*n, (m+2)*n 
  Where m is a positive integer greater than 2.  
• We need to find out the remainder of  \frac{(m+2)*n}{2} 
 o Now,  (\frac{(m+2)*n}{2 })_R= (\frac{m*n}{2})_R + (\frac{2*n}{2})_R 
o  ⟹ (\frac{(m+2)*n}{2} )_R= (\frac{m*n}{2})_R + 0 = (\frac{m*n}{2})_R 
  Where  ()_R  represents “remainder of”   
Therefore, we need to find the remainder when n*m is divided by 2.

Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:  The median of the sequence is even.
 • Median of the sequence M = n*m
• Since  n*m  is even
 o  (\frac{(m+2)*n}{2} )_R= (\frac{m*n}{2})_R = 0   
Hence, statement 1 is sufficient and we can eliminate answer Options B, C and E.

Statement 2:  The second term of the sequence is odd.
 • It means,  (m-1)*n = odd . This is possible only if  (m-1) = odd  and  n = odd .
• If  m- 1 = odd ⟹ m= even .
• Therefore,  (\frac{(m+2)*n}{2} )_R= (\frac{m*n}{2})_R = 0  
Hence, statement 2 is also sufficient and the correct answer is  Option D.

Archit3110 · Answer

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

CrackverbalGMAT · Answer

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!

sambitspm · Answer

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.

Lipun · Answer

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

bumpbot · Answer

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An increasing sequence M consists of 5 consecutive positive multiples

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An increasing sequence M consists of 5 consecutive positive multiples

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