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What is the remainder when a positive integer \(p\) is divided by 2?

Notice that a positive integer can yield only two remainders upon division by 2: 0 (when the dividend is even) and 1 (when the dividend is odd). So, we are basically asked whether \(p\) is even or odd.

(1) \(p\) is a prime integer. \(p\) could be 2, so even, as well as 3 (for example), so odd. Not sufficient.

(2) \(p \gt 10\). Clearly insufficient.

(1)+(2) Any prime number greater than 10 is odd and thus leaves the remainder of 1 when divided by 2. Sufficient.