D01-12

Official Solution:


If \(N = 1234x\), where \(x\) is the units digit of \(N\), is \(N\) a multiple of 5?

In order for \(1234x\) to be divisible by 5, \(x\) must be either 0 or 5. So, the question asks whether \(x\) equals to 0 or 5.

(1) \(x!\) is not divisible by 5.

Since \(x\) is a digit, \(x\) can be 0, 1, 2, 3, or 4 (note that 0!=1). If \(x=0\), \(N\) will be divisible by 5 but if \(x\) is1, 2, 3, or 4, \(N\) will not be divisible by 5. Not sufficient.

(2) \(x\) is divisible by 9.

Since \(x\) is a digit, \(x\) can be 0 or 9 (note that zero is divisible by every integer except zero itself). Not sufficient.

(1)+(2) The only value that satisfies both statements is \(x=0\). Thus, \(N = 12340\), which is divisible by 5. Sufficient.


Answer: C