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rss GMAT Practice Question (One of Hundreds)
What is the remainder of a positive integer N when it is divided by 2?
  1. N contains odd numbers as factors
  2. N is a multiple of 15
Correct Answer: E
The question can be simplified to: is N even? Or: what is the parity of N?
  1. Any positive integer that is divided by 2 will have a remainder of 1 if it is odd. However, it will not have a remainder if it is even.
    N/2 --> Remainder = 0 if N is even
    N/2 --> Remainder = 1 if N is odd
  2. Evaluate Statement (1) alone.
    1. If a number contains only odd factors, it will be odd (and will have a remainder of 1 when divided by 2). If a number contains at least one even factor, it will be even (and divisible by 2).
      15 = 3*5 {only odd factors; not divisible by 2; remainder of 1}
      21 = 3*7 {only odd factors; not divisible by 2; remainder of 1}
      63 = 3*3*7 {only odd factors; not divisible by 2; remainder of 1}

      30 = 3*5*2 {contains an even factor; divisible by 2}
      42 = 3*7*2 {contains an even factor; divisible by 2}
      50 = 5*5*2 {contains an even factor; divisible by 2}
    2. Simply because "N contains odd numbers as factors" does not mean that all of N's factors are odd. Consequently, it is entirely possible that N contains an even factor, in which case N is even and N is divisible by 2. Possible values for N:
      18 = 2*3*3 {contains odd factors, but is divisible by 2; remainder = 0}
      30 = 2*5*3 {contains odd factors, but is divisible by 2; remainder = 0}
      But:
      27 = 3*3*3 {contains odd factors, but is not divisible by 2; remainder = 1}
      15 = 3*5 {contains odd factors, but is not divisible by 2; remainder = 1}
    3. Since some values of N that meet the conditions of Statement (1) are divisible by 2 while other values that also meet the conditions of Statement (1) are not divisible by 2, Statement (1) does not provide sufficient information to definitively determine whether N is divisible by 2.
    4. Statement (1) alone is NOT SUFFICIENT.
  3. Evaluate Statement (2) alone.
    1. Since "N is a multiple of 15", possible values for N include:
      15, 30, 45, 60, 75, 90
    2. Possible values for N give different remainders when divided by 2:
      15/2 --> Remainder = 1
      30/2 --> Remainder = 0
      45/2 --> Remainder = 1
      60/2 --> Remainder = 0
      75/2 --> Remainder = 1
      90/2 --> Remainder = 0
    3. Since different legitimate values of N give different remainders when divided by 2, Statement (2) is not sufficient for determining the remainder when N is divided by 2.
    4. Statement (2) alone is NOT SUFFICIENT.
  4. Evaluate Statements (1) and (2).
    1. Since "N is a multiple of 15" and "N contains odd numbers as factors", possible values for N include:
      15, 30, 45, 60, 75, 90
    2. Adding Statement (1) to Statement (2) does not provide any additional information since any number that is a multiple of 15 must also have odd numbers as factors.
    3. Possible values for N give different remainders when divided by 2:
      15/2 --> Remainder = 1
      30/2 --> Remainder = 0
      45/2 --> Remainder = 1
      60/2 --> Remainder = 0
      75/2 --> Remainder = 1
      90/2 --> Remainder = 0
    4. Since different legitimate values of N give different remainders when divided by 2, Statements (1) and (2) are not sufficient for determining the remainder when N is divided by 2.
    5. Statements (1) and (2), even when taken together, are NOT SUFFICIENT.
  5. Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, answer E is correct.