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GMAT Practice Question (One of Hundreds)
What is the remainder of a positive integer N when it is divided by 2?
N contains odd numbers as factors
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?
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
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}
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}
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.
Since "N is a multiple of 15", possible values for N include:
15, 30, 45, 60, 75, 90
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
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.
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
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.
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
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.
Statements (1) and (2), even when taken together, are NOT SUFFICIENT.
Since Statement (1) alone is NOT SUFFICIENT and Statement (2) alone is NOT SUFFICIENT, answer E is correct.