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# If j and k are positive integers where k > j, what is the

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VP
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If j and k are positive integers where k > j, what is the [#permalink]

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08 Aug 2006, 14:49
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If j and k are positive integers where k > j, what is the value of the remainder when k is divided by j?

(1) There exists a positive integer m such that k = jm + 5.

(2) j > 5
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08 Aug 2006, 14:58
duttsit wrote:
If j and k are positive integers where k > j, what is the value of the remainder when k is divided by j?

(1) There exists a positive integer m such that k = jm + 5.

(2) j > 5

(1) (k-5)/j =m so k-5 is a multiple of j

Thus k is a multiple of j plus 5

if j is greater than 5, 5 is the remainder when k is divided by j
if j is 5 (or less), the remainder will be less than 5 (NOT SUFF)

But with (2), we have an answer!
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09 Aug 2006, 09:56
Need to find r in
k = j x N + r
for some integer N

S1: k = jxm +5

If j < 5, then the equation is true, but the remainder is not 5. Not sufficient.

S2: j > 5
Doesn't give us any information about k.

Not sufficient.

S1 & S2:
If j > 5, then if k=jxm+5, then the remainder must be 5 as the divisor is > 5.
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09 Aug 2006, 10:10
Got C

Given k>j>0

ST1 k = jm+5

Hence k/j =m + 5/j (j can be any value > 0) Hence NOT SUFF

ST2 = j>5

By itself statement 2 is INSUFF

Comnbining the two statements, from 1 k/j = m + 5/j

Hence if j >5 the remainder is always 5 i.e. 5/6 5/10 etc...
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09 Aug 2006, 10:13
C me 2. You guys nailed it.
Have to wait till Moday, though, for OA/OE.
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09 Aug 2006, 13:34
If j and k are positive integers where k > j, what is the value of the remainder when k is divided by j?

(1) There exists a positive integer m such that k = jm + 5.

(2) j > 5

by definition of a remainder.

if x.y and x is not devisible by y not = zero ,there exist an intiger q( qoutient)

such that x/y = qy+r
where 0 < r < y

so WITHOUT KNOWING THE RELATION BETWEEN y and r we can never assume that 5 is a remainder in the above question.

So c must be the answer.
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14 Aug 2006, 10:50
OA C.
(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.

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14 Aug 2006, 13:09
What a nice (though long) explanation!
14 Aug 2006, 13:09
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