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# When the positive integer k is divided by the positive integ

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Re: When the positive integer k is divided by the positive integ [#permalink]
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Hi joaomario,

Here's an approach that's based on Number Properties and a bit of "brute force" math:

We're told that K and N are both INTEGERS.

Since K/N = 81.2, we can say that K = 81.2(N)

N has to "multiply out" the .2 so that K becomes an INTEGER. With the answers that we have to work with, N has to be a multiple of 5. Eliminate A and E.

With the remaining answers, we can TEST THE ANSWERS and find the one that fits the rest of the info (K/N = 81.2 and K/N has a remainder of 11)

Answer B: If N = 20, then K = 1624; 1624/20 has a remainder of 4 NOT A MATCH
Answer C: If N = 55, then K = 4466; 4466/55 has a remainder of 11 MATCH.

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Re: When the positive integer k is divided by the positive integ [#permalink]
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joaomario wrote:
When the positive integer k is divided by the positive integer n , the remainder is 11. If k/n = 81.2 , what is the value of n?

A. 9
B. 20
C. 55
D. 70
E. 81

Can someone break this down for me please?

Here is a response from Beat The GMAT (/tricky-remainder-problem-t270176.html)

Before I past the question, I am confused with how the user came up with K = 81n +11. Where did the .2 go for this representation?

"If, when k is divided by n, the remainder is 11, we could say that some multiple of n plus 11 equals k:
xn + 11 = k

If k/n = 81.2, that means that "some multiple of n" (aka the quotient) is 81, and the remainder is represented by the 0.2.

k = 81.2n

and

k = 81n + 11

Now, we can simply set these expressions equal to each other, since they're both equal to k:

81.2n = 81n + 11
0.2n = 11
n = 55"

If K/N is 81,2 and the remainder is 11 we can say that 11=0,2n
11/0,2=55
N=55

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Re: When the positive integer k is divided by the positive integ [#permalink]
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joaomario wrote:
When the positive integer k is divided by the positive integer n , the remainder is 11. If k/n = 81.2 , what is the value of n?

A. 9
B. 20
C. 55
D. 70
E. 81

We can create two remainder equations:

k/n = 81 + 2/10

k/n = 81 + 1/5

and

k/n = 81 + 11/n

Thus:

1/5 = 11/n

n = 55

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Re: When the positive integer k is divided by the positive integ [#permalink]
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pra1785 wrote:
Bunuel wrote:
joaomario wrote:
When the positive integer k is divided by the positive integer n , the remainder is 11. If k/n = 81.2 , what is the value of n?

A. 9
B. 20
C. 55
D. 70
E. 81

Can someone break this down for me please?

Here is a response from Beat The GMAT (/tricky-remainder-problem-t270176.html)

Before I past the question, I am confused with how the user came up with K = 81n +11. Where did the .2 go for this representation?

"If, when k is divided by n, the remainder is 11, we could say that some multiple of n plus 11 equals k:
xn + 11 = k

If k/n = 81.2, that means that "some multiple of n" (aka the quotient) is 81, and the remainder is represented by the 0.2.

k = 81.2n

and

k = 81n + 11

Now, we can simply set these expressions equal to each other, since they're both equal to k:

81.2n = 81n + 11
0.2n = 11
n = 55"

If $$x$$ and $$y$$ are positive integers, there exist unique integers $$q$$ and $$r$$, called the quotient and remainder, respectively, such that $$y =divisor*quotient+remainder= xq + r$$ and $$0\leq{r}<x$$.

For example, when 15 is divided by 6, the quotient is 2 and the remainder is 3 since $$15 = 6*2 + 3$$.

Hence, the positive integer k is divided by the positive integer n, the remainder is 11, could be written as k = nq + 11. Divide by n: k/n = q + 11/n.

We are also given that k/n = 81.2 = 81 + 0.2. So, the quotient, q, is 81 and 11/n is 0.2: 11/n = 0.2 --> n = 55.

Similar questions to practice:
https://gmatclub.com/forum/when-positive ... 06493.html
https://gmatclub.com/forum/if-s-and-t-ar ... 35190.html

P.S. Please read carefully and follow: https://gmatclub.com/forum/rules-for-pos ... 33935.html Pay attention to rule 3. Thank you.

Hi Bunuel,

"We are also given that k/n = 81.2 = 81 + 0.2. So, the quotient, q, is 81 and 11/n is 0.2: 11/n = 0.2 --> n = 55."

I was wondering, why can't q = 0.2 and 81=11/n ?

The point is that the quotient must be an integer.

For example, 15 divided by 6 gives the quotient of 2 and the remainder of 3: 15 = 2*6 + 3. The quotient (2 in our case) is the greatest whole number of times a divisor (6 in our case) may be subtracted from a dividend (15) without the remainder becoming negative.
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Re: When the positive integer k is divided by the positive integ [#permalink]
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davidbeckham wrote:
could someone please explain the logic behind writing 81.2 in form 81 + 0.2 or 81 + 1/5?

Everything you learn about remainders will be about integers, so you generally would always want to rewrite a question like this using integers instead of decimals. From the definition of a quotient and a remainder, then when we divide integer n by integer d,

n = qd + r

where integer q is the quotient, and integer r is the remainder. The remainder must be between 0 and d-1 inclusive, so r < d is always true. If you divide by d on both sides, you get

n/d = q + (r/d)

Since r < d, then r/d must be between 0 and 1. So when you divide n by d, the quotient q is the whole number part of the result, and r/d is the decimal or fractional part of the result. If we use that for this question (and I'm going to change the letters, because whoever designed this question did something profoundly annoying -- they're using "n" where "d" is normally used in a quotient/remainder definition, and using "k" where "n" would normally be used) :

When the positive integer n is divided by the positive integer d , the remainder is 11. If n/d = 81.2 , what is the value of d?

then 81 is the whole number part of the result, so is the quotient, and 0.2 = 1/5 is the decimal part of the result, so equals r/d. Since r = 11, and r/d = 1/5, we have 11/d = 1/5 and d = 55.

So if you replace 0.2 with a fraction of integers, and you know how to use the definition of a remainder, it's a one-line solution. It's true that you can solve this question in other ways, as the post above mine suggests, but if you're doing something like testing answers here, you're spending far more time on this question than you need to.
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Re: When the positive integer k is divided by the positive integ [#permalink]
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this question is not difficult but ya , a bit tricky to understand when you see this question at first.
so ,
K/N = 11 i.e. K= NQ+11
K/N =81.2= K=N*81+0.2 for example : 16/3= 3*5+1
now ,
K= 81N+11
81.2N=81N+11
N=55.
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Re: When the positive integer k is divided by the positive integ [#permalink]
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We're told $$\frac{k}{n} = 81.2$$

We can rewrite 81.2 as $$81 \frac{1}{5}$$

That 1/5 represents a remainder of 11.

Therefore, 5/5 = 55.

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Re: When the positive integer k is divided by the positive integ [#permalink]
Bunuel wrote:
joaomario wrote:
When the positive integer k is divided by the positive integer n , the remainder is 11. If k/n = 81.2 , what is the value of n?

A. 9
B. 20
C. 55
D. 70
E. 81

Can someone break this down for me please?

Here is a response from Beat The GMAT (/tricky-remainder-problem-t270176.html)

Before I past the question, I am confused with how the user came up with K = 81n +11. Where did the .2 go for this representation?

"If, when k is divided by n, the remainder is 11, we could say that some multiple of n plus 11 equals k:
xn + 11 = k

If k/n = 81.2, that means that "some multiple of n" (aka the quotient) is 81, and the remainder is represented by the 0.2.

k = 81.2n

and

k = 81n + 11

Now, we can simply set these expressions equal to each other, since they're both equal to k:

81.2n = 81n + 11
0.2n = 11
n = 55"

If $$x$$ and $$y$$ are positive integers, there exist unique integers $$q$$ and $$r$$, called the quotient and remainder, respectively, such that $$y =divisor*quotient+remainder= xq + r$$ and $$0\leq{r}<x$$.

For example, when 15 is divided by 6, the quotient is 2 and the remainder is 3 since $$15 = 6*2 + 3$$.

Hence, the positive integer k is divided by the positive integer n, the remainder is 11, could be written as k = nq + 11. Divide by n: k/n = q + 11/n.

We are also given that k/n = 81.2 = 81 + 0.2. So, the quotient, q, is 81 and 11/n is 0.2: 11/n = 0.2 --> n = 55.

Similar questions to practice:
https://gmatclub.com/forum/when-positive ... 06493.html
https://gmatclub.com/forum/if-s-and-t-ar ... 35190.html

Theory on remainders problems: https://gmatclub.com/forum/remainders-144665.html
Tips on Remainders: https://gmatclub.com/forum/remainders-ti ... s#p1376126

Units digits, exponents, remainders problems: https://gmatclub.com/forum/new-units-dig ... 68569.html

All DS remainders problems to practice: https://gmatclub.com/forum/search.php?se ... tag_id=198
All PS remainders problems to practice: https://gmatclub.com/forum/search.php?se ... tag_id=199

P.S. Please read carefully and follow: https://gmatclub.com/forum/rules-for-pos ... 33935.html Pay attention to rule 3. Thank you.

Hi Bunuel,

"We are also given that k/n = 81.2 = 81 + 0.2. So, the quotient, q, is 81 and 11/n is 0.2: 11/n = 0.2 --> n = 55."

I was wondering, why can't q = 0.2 and 81=11/n ?
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Re: When the positive integer k is divided by the positive integ [#permalink]
Quote:
We are also given that k/n = 81.2 = 81 + 0.2.

could someone please explain the logic behind writing 81.2 in form 81 + 0.2 or 81 + 1/5?
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Re: When the positive integer k is divided by the positive integ [#permalink]
davidbeckham wrote:
Quote:
We are also given that k/n = 81.2 = 81 + 0.2.

could someone please explain the logic behind writing 81.2 in form 81 + 0.2 or 81 + 1/5?

Hi davidbeckham,

GMAT questions can almost always be solved in more than one way (including how you might choose to work through the 'math steps' involved). Sometimes 'converting' information from one format to another can make it easier for you to answer the given question (depending on how you like to "see" your math work).

Some of the explanations in this thread 'convert' 81.2 in the way that you describe, but that is NOT a necessary step to answering this question. Working in that way is just one option for dealing with the fact that both K and N must be INTEGERS and K/N has a remainder of 11.

GMAT assassins aren't born, they're made,
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Re: When the positive integer k is divided by the positive integ [#permalink]
Partly a bit of fun. Hope it helps someone out there! This tutor is going old-school for this! (Gosh, his writing is terrible すみません!) *technically this topic was done yesterday - It's now past midnight where I am!

https://ibb.co/2K4Pckp
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Re: When the positive integer k is divided by the positive integ [#permalink]
Video solution from Quant Reasoning starts at 5:45:
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1
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Re: When the positive integer k is divided by the positive integ [#permalink]
basshead How did you get 5/5 is 55? I got the 1/5 is 11, but how the 5/5? I'm a bit lost & slow lol
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Re: When the positive integer k is divided by the positive integ [#permalink]
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