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(1) INSUFFICIENT: At first glance, this may seem sufficient since if 5 is the remainder when k is divided by j, then there will always exist a positive integer m such that k = jm + 5. In this case, m is equal to the integer quotient and 5 is the remainder. For example, if k = 13 and j = 8, and 13 divided by 8 has remainder 5, it must follow that there exists an m such that k = jm + 5: m = 1 and 13 = (8)(1) + 5.
However, the logic does not go the other way: 5 is not necessarily the remainder when k is divided by j. For example, if k = 13 and j = 2, there exists an m (m = 4) such that k = jm + 5: 13 = (2)(4) + 5, consistent with statement (1), yet 13 divided by 2 has remainder 1 rather than 5.
When j < 5 (e.g., 2 < 5); this means that j can go into 5 (e.g., 2 can go into 5) at least one more time, and consequently m is not the true quotient of k divided by j and 5 is not the true remainder. Similarly, if we let k = 14 and j = 3, there exists an m (e.g., m = 3) such that statement (1) is also satisfied [i.e., 14 = (3)(3) + 5], yet the remainder when 14 is divided by 3 is 2, a different result than the first example.
Statement (1) tells us that k = jm + 5, where m is a positive integer. That means that k/j = m + 5/j = integer + 5/j. Thus, the remainder when k is divided by j is either 5 (when j > 5), or equal to the remainder of 5/j (when j is 5 or less). Since we do not know whether j is greater than or less than 5, we cannot determine the remainder when k is divided by j.
(2) INSUFFICIENT: This only gives the range of possible values of j and by itself does not give any insight as to the value of the remainder when k is divided by j.
(1) AND (2) SUFFICIENT: Statement (1) was not sufficient because we were not given whether 5 > j, so we could not be sure whether j could go into 5 (or k) any additional times. However, (2) tells us that j > 5, so we now know that j cannot go into 5 any more times. This means that m is the exact number of times that k can be divided by j and that 5 is the true remainder.
Another way of putting this is: From statement (1) we know that k/j = m + 5/j = integer + 5/j. From statement (2) we know that j > 5. Therefore, the remainder when k is divided by j must always be 5.
The correct answer is C.
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