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

Positive integer \(a\) divided by positive integer \(d\) yields a reminder of \(r\) can always be expressed as \(a=qd+r\), where \(q\) is called a quotient and \(r\) is called a remainder, note here that \(0\leq{r}<d\) (remainder is non-negative integer and always less than divisor).

So according to above \(k\) is divided by \(j\) yields a remainder of \(r\) can be expressed as: \(k=qj+r\), where \(0\leq{r}<j=divisor\). Question: \(r=?\)

(1) There exists a positive integer m such that k = jm + 5 --> it's tempting to say that this statement is sufficient and \(r=5\), as given equation is very similar to \(k=qj+r\). But we don't know whether \(5<j\): remainder must be less than divisor.

For example:
If \(k=6\) and \(j=1\) then \(6=1*1+5\) and the remainder upon division 6 by 1 is zero;
If \(k=11\) and \(j=6\) then \(11=1*6+5\) and the remainder upon division 11 by 6 is 5.
Not sufficient.

(2) \(j > 5\) --> clearly insufficient.

(1)+(2) \(k = jm + 5\) and \(j > 5\) --> direct formula of remainder as defined above --> \(r=5\). Sufficient.

Or: \(k = jm + 5\) --> first term \(jm\) is clearly divisible by \(j\) and 5 divided by \(j\) as (\(j>5\)) yields remainder of 5.

Answer: C.
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Ans: C

Statement 1: k=jm+5
This is of the form "Quotient x J + Remainder". However J could be 2, 3, 4, in which case the remainder would not be 5.

Statement 2: j>5
Insufficient. Just the value of J is not sufficient to find what the remainder is.

Combining both the equations we get that the remainder is 5.

Thanks for the gr8 explanation !!

Hi Bunuel,

Could you please elaborate as to why A is not the right answer. Would really appreciate it. Thanks

Consider the examples for the first statement given in my solution proving that this statement is not sufficient.
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Very tricky. Nice question! As always, great explanation Bunuel!!

I do not understand why the reminder is still 5....

If you have 25=20*2 + 5 than reminder is 5. But if K/J than the reminder is 5/10: 0.5 not 5. And indeed 25/10=2.5 and 2+0.5=2.5

Therefore, the value of the reminder when K is divided by J is correlated with the value of J.

This is why I answered E because we do not know the value of J.

Where did I get wrong?

Thanks!
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The remainder when k=25 is divided by j=20 is 5.
The remainder when k=5 is divided by j=10 is 5 too.

Hope it's clear.
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Hi Bunnel,

If 2) j<5, will the Answer be E?

Thank you

Yes, that's correct.
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1) K = jm + 5 -> K/j = m + 5/j -> remainder of 5/j is the remainder, without knowing J value remainder could be anything -> insufficient

2) j>5 remainder could be anything - insufficient

(1)(2) if J>5 remainder of 5/j is 5 -> sufficient

Answer C

Hi, would be grateful if someone could elaborate on first statement. Can't understand how given statement 'k=jm+5' is not the same as 'a=qd+r'.

Thanks

Have you read this:
(1) There exists a positive integer m such that k = jm + 5 --> it's tempting to say that this statement is sufficient and \(r=5\), as given equation is very similar to \(k=qj+r\). But we don't know whether \(5<j\): remainder must be less than divisor.

For example:
If \(k=6\) and \(j=1\) then \(6=1*1+5\) and the remainder upon division 6 by 1 is zero;
If \(k=11\) and \(j=6\) then \(11=1*6+5\) and the remainder upon division 11 by 6 is 5.
Not sufficient.
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Stmt 1: we need to findd k/j so as per the stmt 1 jm+5/j
This gives us m + 5/j
As m is an integer we need to find the remainder for 5/j
Not suff

Stmt 2: j>5 does not tell us anything. So insuff

Combining we get
J>5 so 5/j will always give a remainder of 5

So the ans is C

We are told that k > j > 0, and both k and j are integers. The remainder when k is divided by j may be expressed as r in this formula:
k = jq + r
In this formula,
(a) all of the variables are integers,
(b) q (the quotient) is the greatest number of j's such that jx < k, and
(c) r < j.
(If r were greater than j, then q would not be the greatest number of j's in k.)
Thus, the question may be rephrased: “If k = jq + r, and q is maximized such that jq < k and r < j,
what is the value of r?”
(1) INSUFFICIENT: At first glance, this may seem sufficient since it is in the form of our remainder equation. Certainly, m could equal q (the quotient) and r (the remainder) could be 5.

For example, k = 13 and j = 8 yield a remainder of 5 when k is divided by j: 13 = (8)(1) + 5, where m = 1 is the greatest number of 8's such that (8)(1) < 13, and r < j (i.e. 5 < 8).
However, this statement does not indicate whether m is the greatest number of j's such that jm < k and r < j, as our rephrased question requires.
For example, k = 13 and j = 2 may be expressed in this form: 13 = (2)(4) + 5, where m = 4.
However, 5 is not the remainder because 5 > j, and 4 is not the greatest number of 2's in 13. When 13 is divided by 2, the quotient is 6 and the remainder is 1.
If j ≤ 5, then 5 cannot be the remainder and m is not the quotient.
If j > 5, then 5 must be the remainder and m must be the quotient.

(2) INSUFFICIENT: This statement gives us a range of possible values of j. Without information about k, we cannot determine anything about the remainder when k is divided by j.

(1) AND (2) SUFFICIENT: Statement (2) tells us that j > 5, so we can conclude from statement (1) that 5 is the remainder and m is the quotient when k is divided by j.
The correct answer is C.

Prompt analysi
k and j are positive integers such that k>j

Superset
The answer will be a positive integer

Translation
In order to find the answer, we need:
1# exact value of x and y
2# any relation between x and y
3# some property of x and y

Statement analysis
St 1: k =jm +5 . we can say the 5 is the remainder only if j>5. since there is no such condition give the statement is INSUFFICIENT

St 2: j>5. Cannot be said anything about the remainder. INSUFFICIENT

St 1 & St 2: k = jm +5 and j>5. we can say that 5 is the remainder.ANSWER

Option C

I did a much more brute way:

Statement 1:
\(k=mj+5\)

Consider the case of m=any integer, j=1. In this case, the remainder will always be 0.
Consider the case of m=any, j!=1. In this case, the remainder will not always be 0.

INS.

Statement 2:

Clearly INS.

Both:
We know that j>5.
Lets test a few cases:
if m=1, j=6 k=11, and k/j has R5
if m=1, j=7, k=12, k/j has R5
if m=2, j=6, k=17, 17/6 is R5.

So there is a clear pattern established - the remainder is constantly 5.

C is the correct answer

Such a simple thing but still so tricky!!

We routinely write divisor and dividend in the form Dividend = Quotient*Divisor + Remainder
But we know that the remainder must always be less than the divisor (discussed in the link below)

(1) There exists a positive integer m such that k = jm + 5.
k = jm + 5
seems to be in the format Dividend = Quotient*Divisor + Remainder so it may prompt us to think that the remainder is 5 but note that this is true only if the divisor 'j' is greater than 5.
Say if j = 1, we can write many numbers k as 8 = 3*1 + 5 or 12 = 7*1 + 5 etc. But here when k is divided by j, the remainder is actually 1.
Hence this is not sufficient alone.

(2) j > 5
No info on k. Not sufficient alone.

Both Together, we know that k = jm + 5 and j > 5 so then it makes sense that when we divide k by j, the remainder is 5.

Answer (C)

Check this video on Division and Remainders: https://youtu.be/A5abKfUBFSc