If n is a positive integer and r is the remainder when 4 + 7n is divid

If n is a positive integer, and r is the remainder when 4 + 7n is divided by 3, what is the value of r?

r is the remainder when 4+7n is divided by 3 --> \(4+7n=3q+r\), where \(r\) is an integer \(0\leq{r}<3\). \(r=?\)

(1) n+1 is divisible by 3 --> \(n+1=3k\), or \(n=3k-1\) --> \(4+7(3k-1)=3q+r\) --> \(3(7k-1-q)=r\) --> so \(r\) is multiple of 3, but it's an integer in the range \(0\leq{r}<3\). Only multiple of 3 in this range is 0 --> \(r=0\). Sufficient.

(2) n>20. Clearly not sufficient. \(n=21\), \(4+7n=151=3q+r\), \(r=1\) BUT \(n=22\), \(4+7n=158=3q+r\), \(r=2\). Not sufficient.

Answer: A.
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7n+4=3(2n+1) + n+1

So remainder when divided by 3 will be same as remainder left by n+1

1) sufficient ... Gives the answer
2) insufficient ... Irrelevant. Eg n= 22,23,24 all are possible

Answer is (a)
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i think i may have an easier way....
s1) 7n+4 = (6n+3)+(n+1)
if (n+1)/3 = an integer, so must 3 times (n+1)....which is (6n+3)
s2) Obviously NS

Little correction: 3 times (n+1) is 3n+3 not (6n+3).

But you are right, we can solve with this approach as well:

(1) n+1 is divisible by 3 --> \(7n+4=(4n+4)+3n=4(n+1)+3n\) --> \(4(n+1)\) is divisible by 3 as \(n+1\) is, and \(3n\) is obviously divisible by 3 as it has 3 as multiple, thus their sum, \(7n+4\), is also divisible by 3, which means that remainder upon division \(7n+4\) by 3 will be 0. Sufficient.

Hope it's clear.
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Can you explain how you get to 7n+4=3(2n+1)?

The information in the statement A is used favorably to tweak the equation in the question.

Hence 7n+4 becomes 3(2n+1) + (n+1). Now since 3(2n+1) leaves a remainder 0 when divided by 3, the remainder of 7n+4 will be the same as the remainder of (n+1).

Since (n+1) is also given in option A to be divisible by 3, hence remainder 0. Statement A is sufficient.

Just trying to split it out into parts divisible by 3
7n becomes 6n+n
4 becomes 3+1
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statmnt 1:
n+1 ==>div by 3
therefore 7(n+1) ==>div. by 3
(7n+4)+3 ==>div by ===>this means 7n+4 is div by 3
sufficient

statement 2:
n>20
let n=30==>7n+4=214==>when divided by 3 remainder i s1
let n=40==>7n+4=284==>when divided by 3 remainder i s2
hence not sufficient

hence A

Q. What is r?

(1). (n+1) div by 3

By question stem we know that

4 + 7n = 3Q + r

Splitting the equation as below

4+4n+3n = 3Q + r

=> 3n + 4 (n+1) = 3Q + r

LHS is divisible by 3 as (n+1) is div by 3, so RHS should also be divisible by 3 hence r should be 0

(2).

4 + 7n = 3Q +r

Case 1: n=21

4 + 7*21 = 3Q +r

LHS gives 4 as remainder when divided by 3 so r=4

Case 2: n=22

4 + 7*22 = 3Q + r

No info about r, hence inconsistent

(A) it is!

I am not sure if it has been asked/discussed before, but can we use the following approach:

re-write 4 + 7n as (3+1) + (6n + 1). now if we divide this by 3 we are left with (1) + (2n +1), which is essentially (2n + 2) ------> 2(n+1)/3 leaves no remainder or in other words 0 - using statement 1 information.

Please correct me if I this approach is incorrect.

4 + 7n = (3+1) + (6n + n) not (3+1) + (6n + 1).

You can solve (1) in another way: \(4 + 7n = 4 + 4n + 3n = 4(n + 1) + 3n\). First statement says that \(n + 1\) is is divisible by 3, thus \(4(n + 1) + 3n = (a \ multiple \ of \ 3) + (a \ multiple \ of \ 3)\). Therefore \(4 + 7n\) yields the remainder of 0, when divided by 3.

Hope it helps.
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1) n+1 is a multiple of 3.

If n=2 (we're allowed to pick 2 since 2+1 is a multiple of 3), then (4+14)/3 = 18/3 = 6rem0
If n=5 (we're allowed to pick 5 since 5+1 is a multiple of 3), then (4+35)/3 = 39/3 = 13rem0

at this point you might already be conviced that you'll always get the same answer, but we could try one more just to be safe:

If n=8 (we're allowed to pick 8 since 8+1 is a multiple of 3), then (4+56) = 60/3 = 20rem0

For all 3 plug-ins we get r=0.. sufficient!

2) n > 20

If n=21, then (4+147)/3 = 151/3 = 50rem1
Insuff.

Hence A.
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Remainder of (4 + 7n) when divided by 3 can be conceptually calculated in the following way. We must remember that the remainder is always less than the divisor and hence we should reduce the final calculated quantity so that it becomes less than the divisor to obtain the "true" remainder.

"Remainder" of (4 + 7n) divided by 3 = (Remainder of 4 divided by 3) + (Remainder of 7 divided by 3)*(Remainder of n divided by 3); Then compare the obtained quantity to 3. Since remainder is always less than the divisor (3), we need to further reduce the obtained quantity through division by 3.

So we have:

"Remainder" of (4 + 7n) divided by 3 = (1) +(1)*(Remainder of n divided by 3) and keep dividing by 3 to obtain the true remainder less than the divisor (3).


(1) n + 1 is divisible by 3

This tells us that when n is divided by 3, the remainder is 2.

From above reasoning, we have:

"Remainder" of (4 + 7n) divided by 3 = (1) +(1)*(2) = 3, and now we need to divide further by 3. This process gives us a value of zero.

The "true" remainder is equal to zero.

SUFFICIENT

(2) n > 20.

This is clearly insufficient. Try n = 21 and n = 22. The respective remainders of (4+7n) when divided by 3 will be 1 and 2.

ANSWER: (A)

Hi Bunuel

I think there seems to be a typo in the statement highlighted. It should be 4 + 7n and not 4n + 7 as per the question and other statements.

Please let me know if otherwise.

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Edited the typo. Thank you!
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I proved statement 1 as sufficient in a bit of a different way. I learned this in school, so it may not be known to many (but it may help some).
If n + 1 is divisible by 3, then: n + 1 = 0 mod 3
subtract 1 on both sides
n = -1 mod 3
n = 2 mod 3

Now we know that n is congruent to 2 mod 3. Let us plug 2 in for n in the original equation.

4 + 7n = r mod 3
4 + 7 * 2 = r mod 3
18 = r mod 3

18 mod 3 = 0, so therefore r = 0
suff.

Info from the stem:
1. N > 0 and integer
2. 7N + 4 when divided by 3 leaves a remainder R

7N + 4 can be re-written as 6N + 3 + (N + 1). 6N and 3 are divisible by 3 always, so we need to find what is the remainder when N+1 is divided by 3.

S1: N+1 is divisible by 3 - implies R=0 (unique answer). Sufficient.
S2: N > 20 - multiple answers are possible. Insufficient.

Hence the correct answer is option A.

Video solution from Quant Reasoning:
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If n is a positive integer and r is the remainder when 4 + 7n is divided by 3, what is the value of r ?

(1) n + 1 is divisible by 3.
(2) n > 20

Statement 1 -
4 + 7n can be rewritten as 7n + 4 and that can be split as 7n + 7 - 3.
7n + 7 - 3 = 7(n + 1) - 3

We know, n + 1 is divisible by 3 & 3 is also divisible by 3. So r = 0. Statement 1 is sufficient.

Answer choices to pick from -> A & D (AD/BCE rule)

Now we just need to check if statement 2 by itself.

Statement 2 -
n > 20

n can take any value such as 21, 22, 23, etc. Each value gives us a different number when plugged into 4 + 7n, and thus a different remainder when divided by 3. Hence, not sufficient.

Thus, answer is option A.