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# When the number 777 is divided by the integer N, the remaind

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When the number 777 is divided by the integer N, the remaind [#permalink]  15 Sep 2010, 13:04
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When the number 777 is divided by the integer N, the remainder is 77. How many integer possibilities are there for N?
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Re: Remainder [#permalink]  15 Sep 2010, 13:20
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mainhoon wrote:
When the number 777 is divided by the integer N, the remainder is 77. How many integer possibilities are there for N?

I think it should me mentioned that $$n$$ is a positive integer.

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 we'd have: $$777=qn+77$$, where $$remainder=77<n=divisor$$ --> $$qn=700=2^2*5^2*7$$ --> as $$n$$ must be more than 77 then $$n$$ could take only 5 values: 100, 140, 175, 350, and 700.

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Re: Remainder [#permalink]  15 Sep 2010, 13:39
Can N be allowed to be negative? Does GMAT allow that possibility and if so, how does one answer this then?
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Re: Remainder [#permalink]  15 Sep 2010, 13:42
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mainhoon wrote:
Can N be allowed to be negative? Does GMAT allow that possibility and if so, how does one answer this then?

No, it can not be the case, at least for GMAT. Every GMAT divisibility question will tell you in advance that any unknowns represent positive integers.

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Re: Remainder [#permalink]  15 Sep 2010, 13:49
mainhoon wrote:
Can N be allowed to be negative? Does GMAT allow that possibility and if so, how does one answer this then?

the problem with negative numbers is that there is no unique definition of remainder

the only condition is that abs(remainder)<abs(divisor)

But even if we follow that, it is enough to tell us that the possible divisors is just double. All the positive ones listed above as well as -1*those numbers
hence, 10
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Re: Remainder [#permalink]  18 Oct 2010, 10:29
Bunuel wrote:
mainhoon wrote:
When the number 777 is divided by the integer N, the remainder is 77. How many integer possibilities are there for N?

I think it should me mentioned that $$n$$ is a positive integer.

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 we'd have: $$777=qn+77$$, where $$remainder=77<n=divisor$$ --> $$qn=700=2^2*5^2*7$$ --> as $$n$$ must be more than 77 then $$n$$ could take only 5 values: 100, 140, 175, 350, and 700.

I couldn't understand how we arrived at 100,140,175,350 and 700... is it something that we did manually or erupted out of the calculation given here
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Re: Remainder [#permalink]  19 Oct 2010, 01:30
hirendhanak wrote:
Bunuel wrote:
mainhoon wrote:
When the number 777 is divided by the integer N, the remainder is 77. How many integer possibilities are there for N?

I think it should me mentioned that $$n$$ is a positive integer.

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 we'd have: $$777=qn+77$$, where $$remainder=77<n=divisor$$ --> $$qn=700=2^2*5^2*7$$ --> as $$n$$ must be more than 77 then $$n$$ could take only 5 values: 100, 140, 175, 350, and 700.

I couldn't understand how we arrived at 100,140,175,350 and 700... is it something that we did manually or erupted out of the calculation given here

by multiplying the factors of $$700$$ with each other and selecting only those numbers which result in $$\geq 77$$ till $$700$$.
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Re: Remainder [#permalink]  22 May 2011, 22:25
AtifS wrote:

I couldn't understand how we arrived at 100,140,175,350 and 700... is it something that we did manually or erupted out of the calculation given here

by multiplying the factors of $$700$$ with each other and selecting only those numbers which result in $$\geq 77$$ till $$700$$.[/quote]

whats the fastest way of finding factors of 700
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Re: Remainder [#permalink]  23 May 2011, 06:20
Just break 700 into prime factors, take 5 and 2 as there is a 0 at end. Take 7 also, as the number is 700.

700 = 2^2 * 5^2 * 7
Then the number of factors will be (2+1) * (2+1) * (1+1)

Also, you can refer to Math Book for more details on prime factorization and number of factors.
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Re: Remainder [#permalink]  23 May 2011, 17:35
700 = 2^2 * 5^2 * 7
How to come quickly with divisors greater than 77 from above expression? It took almost two minutes to me.
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Re: Remainder [#permalink]  23 May 2011, 20:49
good concept of number > 77 here.

777-77 = nQ
2^2 * 5^2 * 7 = 700

gives 100,140,175,350 and 700 .
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Re: Remainder [#permalink]  29 Jun 2013, 11:09
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hirendhanak wrote:
Bunuel wrote:
mainhoon wrote:
When the number 777 is divided by the integer N, the remainder is 77. How many integer possibilities are there for N?

I think it should me mentioned that $$n$$ is a positive integer.

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 we'd have: $$777=qn+77$$, where $$remainder=77<n=divisor$$ --> $$qn=700=2^2*5^2*7$$ --> as $$n$$ must be more than 77 then $$n$$ could take only 5 values: 100, 140, 175, 350, and 700.

I couldn't understand how we arrived at 100,140,175,350 and 700... is it something that we did manually or erupted out of the calculation given here

You know that the feasible factors of 700 must be in a range above 77. Thus start breaking down 700 "from the top":

700 x 1
350 x 2
175 x 4
140 x 5
100 x 7

The next one, 70 x 10, is already out of range. That gives you 5 factors.
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Re: When the number 777 is divided by the integer N, the remaind [#permalink]  17 Oct 2013, 17:38
mainhoon wrote:
When the number 777 is divided by the integer N, the remainder is 77. How many integer possibilities are there for N?

Hey there, are there at least some answer choices for this question?
Much appreciated

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Re: When the number 777 is divided by the integer N, the remaind [#permalink]  13 Nov 2014, 15:08
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