LUANDATO wrote:If a, b and c are positive integers and c is odd, are both a and b divisible by c ?
(1) a + b is divisible by c.
(2) a - b is divisible by c.
Statement 1: a+b is divisible by c
Case 1: a=1, b=1, c=1, with the result that a+b=2 is divisible by c=1
In this case, a and b are both divisible by c.
Case 2: a=2, b=1, c=3, with the result that a+b=3 is divisible by c=3
In this case, a and b are NOT both divisible by c.
Since the answer to the question stem is YES in Case 1 but NO in Case 2, INSUFFICIENT.
Statement 2: a-b is divisible by c
Case 1: a=1, b=1, c=1, with the result that a-b=0 is divisible by c=1
In this case, a and b are both divisible by c.
Case 2: a=4, b=1, c=3, with the result that a-b=3 is divisible by c=3
In this case, a and b are NOT both divisible by c.
Since the answer to the question stem is YES in Case 1 but NO in Case 2, INSUFFICIENT.
Statements combined:
Case 1: c=3
Let a+b = 9 and a-b = 3, with the result that a+b and a-b are both divisible by c=3.
Adding together a+b = 9 and a-b = 3, we get:
(a+b) + (a-b) = 9 + 3
2a = 12
a = 6.
Since a=6 and a+b = 9, b=3.
In this case, both a=6 and b=3 are divisible by c=3.
Case 2: c=5
Let a+b = 45 and a-b = 15, with the result that a+b and a-b are both divisible by c=5.
Adding together a+b = 45 and a-b = 15, we get:
(a+b) + (a-b) = 45 + 15
2a = 60
a = 30.
Since a=30 and a+b = 45, b=15.
In this case, both a=30 and b=15 are divisible by c=5.
Case 3: c=7
Let a+b = 105 and a-b = 63, with the result that a+b and a-b are both divisible by c=7.
Adding together a+b = 105 and a-b = 63, we get:
(a+b) + (a-b) = 105 + 63
2a = 168
a = 84.
Since a=84 and a+b = 105, b=21.
In this case, both a=84 and b=21 are divisible by c=7.
In every case, a and b are both divisible by c, with the result that the answer to the question stem is YES.
SUFFICIENT.
The correct answer is
C.
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