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What is the remainder when (47)(49) is divided by 8?

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What is the remainder when (47)(49) is divided by 8? [#permalink]

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New post 11 Jun 2015, 03:55
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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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New post 11 Jun 2015, 04:38
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we can make use of the rule :
Remainder of { (a * b)/n}} = Remainder of (a/n) * Remainder of (b/n)

Here
Remainder of { 47 * 49)/8}} = Remainder of (47/8) * Remainder of (49/8) = 7 * 1 = 7

Answer : E
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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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Bunuel wrote:
What is the remainder when 47*49 is divided by 8?

(A) 1
(B) 3
(C) 4
(D) 5
(E) 7

Kudos for a correct solution.


CONCEPT: Remainder Theorem states that Remainders can be calculated for each term multiplied or added together in Numerator and then the operation (multiplication or addition) can be performed on remainders thereafter and finally remainder can be calculated

i.e. Remainder when 47*49 is divided by 8 = Remainder [47/8] * Remainder [49/8]

i.e. Remainder when 47*49 is divided by 8 = 7 * 1 = 7


Answer: Option
[Reveal] Spoiler:
E

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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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One way to solve this

(8x6-1)(8x6+1)/8
remaninder -1x1=-1 ie 7

Another way

(48-1)(48+1)/8
(48^2-1^2)/8
remainder when 48^2/8 is 0
remainder when -1^2/8 is -1 ie 7

Ans is E

jimmy

Bunuel wrote:
What is the remainder when 47*49 is divided by 8?

(A) 1
(B) 3
(C) 4
(D) 5
(E) 7

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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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New post 13 Jun 2015, 07:53
I think we could also use the last digits of the numbers like this:

47 * 49 = 7*9 = 63.
63/8 = 7,... and leaves a remainder of 7.




One question on one of the solutions above:
Another way

(48-1)(48+1)/8
(48^2-1^2)/8
remainder when 48^2/8 is 0
remainder when -1^2/8 is -1 ie 7

Why does this tell us that the remainder is 7? I read the post about remainder tricks and tips, but it is not that clear to me...
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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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New post 13 Jun 2015, 08:06
pacifist85 wrote:
I think we could also use the last digits of the numbers like this:

47 * 49 = 7*9 = 63.
63/8 = 7,... and leaves a remainder of 7.




One question on one of the solutions above:
Another way

(48-1)(48+1)/8
(48^2-1^2)/8
remainder when 48^2/8 is 0
remainder when -1^2/8 is -1 ie 7

Why does this tell us that the remainder is 7? I read the post about remainder tricks and tips, but it is not that clear to me...


There are two things for you to note

1) the method of calculating remainder by calculating unit digits and dividing by divisor is fundamentally wrong if the divisors are not 2 or 5 or 10.
Just luckily the answer is matching so please don't use this method because it fits to one question by chance.

2) remainder can be written in two forms either positive or negative. The understanding of positive and negative remainders are as follows

Remainder of 7 when a number divided by 8 means the number has 7 extra else the no. Would have been divisible by 8

Remainder of -1 when a number divided by 8 means the number is short by 1 else the no. Would have been divisible by 8

So technically they are sane and remainder of -1 is same as remainder of 7 for a divisor 8

Similarly, remainder of -2 is same as remainder of 6 for a divisor 8

Similarly, remainder of -4 is same as remainder of 5 for a divisor 9

Similarly, remainder of -5 is same as remainder of 7 for a divisor 12

I hope it clears your doubt!
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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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New post 13 Jun 2015, 08:27
Hello,

Thank you for the explanations.

OK I can now understand what remainders for a divisor mean.
However, I still do not understand why when you divide -1^2 by 8 you get a remainder of 7. Isn't it the same as 1/8? This is the 0.125 and I cannot see how you end up with 7 as a remainder..
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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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pacifist85 wrote:
Hello,

Thank you for the explanations.

OK I can now understand what remainders for a divisor mean.
However, I still do not understand why when you divide -1^2 by 8 you get a remainder of 7. Isn't it the same as 1/8? This is the 0.125 and I cannot see how you end up with 7 as a remainder..


Point 1: You are making a mistake again and to identify check the explanation below

Remainder [(47 x 49)/8] = Remainder [(48-1)(48+1)/8] = Remainder [(48^2 - 1^2)/8]
i.e. Remainder [(47 x 49)/8] = Remainder [48^2/8] - Remainder [1^2/8] = Remainder [0] - Remainder [1] = 0-1 = -1

Remainder (-1) means Remainder (7) for Divisor = 8

Please Note: \(-1^2\) is not same as \((-1)^2\)

\(-1^2\) = \(-1\) which is the case here AND \((-1)^2\) = \(+1\)

Point 2: The tradition of saying "Thank you" here on GMAT CLUB is by pressingthe button "+1Kudos" :P :wink:
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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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New post 13 Jun 2015, 10:08
Bunuel wrote:
What is the remainder when 47*49 is divided by 8?

(A) 1
(B) 3
(C) 4
(D) 5
(E) 7

Kudos for a correct solution.


Since the product can be easily calculated here, 47*49=2303
Simply divide this by 8
Remainder=7
Answer E
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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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Further to what GMATinsight wrote:

Please note following(in other words):

1. When you divide an integer by a positive integer N, the possible remainders range from 0 to (N- 1). There are thus N possible remainders.

2. You can add and subtract remainders directly, as long as you correct excess or negative remainders. so to correct the negative remainder, which is unacceptable remainder, add and extra 8. you get 7 as the answer.

Please note here that excess and negative remainders are not acceptable. for more, please refer chapter 10 of manhattan math book 1.

regards

jimmy

GMATinsight wrote:
pacifist85 wrote:
I think we could also use the last digits of the numbers like this:

47 * 49 = 7*9 = 63.
63/8 = 7,... and leaves a remainder of 7.




One question on one of the solutions above:
Another way

(48-1)(48+1)/8
(48^2-1^2)/8
remainder when 48^2/8 is 0
remainder when -1^2/8 is -1 ie 7

Why does this tell us that the remainder is 7? I read the post about remainder tricks and tips, but it is not that clear to me...


There are two things for you to note

1) the method of calculating remainder by calculating unit digits and dividing by divisor is fundamentally wrong if the divisors are not 2 or 5 or 10.
Just luckily the answer is matching so please don't use this method because it fits to one question by chance.

2) remainder can be written in two forms either positive or negative. The understanding of positive and negative remainders are as follows

Remainder of 7 when a number divided by 8 means the number has 7 extra else the no. Would have been divisible by 8

Remainder of -1 when a number divided by 8 means the number is short by 1 else the no. Would have been divisible by 8

So technically they are sane and remainder of -1 is same as remainder of 7 for a divisor 8

Similarly, remainder of -2 is same as remainder of 6 for a divisor 8

Similarly, remainder of -4 is same as remainder of 5 for a divisor 9

Similarly, remainder of -5 is same as remainder of 7 for a divisor 12

I hope it clears your doubt!
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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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47/8=R(-1)
49/8=R(+1)
-1*+1=-1
R(-1) OR R(7)
Answer E
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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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New post 15 Jun 2015, 04:39
Bunuel wrote:
What is the remainder when 47*49 is divided by 8?

(A) 1
(B) 3
(C) 4
(D) 5
(E) 7

Kudos for a correct solution.


MANHATTAN GMAT OFFICIAL SOLUTION:

One way to solve would be to multiply (47)(49), then either divide the result by 8 or repeatedly subtract known multiples of 8 from the result until we are left with a remainder smaller than 8.

An alternative is to rewrite the given product as an equivalent yet easier-to-manipulate product. Note that 47 and 49 are equidistant from 48, a multiple of 8. We can write each of the original factors as terms in the form (a + b) or (a – b).

(47)(49) = (48 + 1)(48 – 1)

Recognizing the “difference of two squares” special product, (a + b)(a – b) = a^2 – b^2, we can quickly manipulate again:

(48 + 1)(48 – 1) = (48^2 – 1^2)

48 is a multiple of 8, and therefore so is 48^2. Thus, (48^2 – 1^2) is 1 less than a multiple of 8. All such numbers (e.g. 7, 15, 23, 31, etc.) have a remainder of 7 when divided by 8.

The correct answer is E.
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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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Because we know that 8 divides into 48 evenly, we just need to find the remainder for 47/8:

49 x 47 = (48 x 47) + 47. We know that 8 divides into 48 x 47 evenly because it divides into 48 evenly so you can remove this portion entirely.
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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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New post 15 Jun 2015, 21:06
Bunuel wrote:
What is the remainder when 47*49 is divided by 8?

(A) 1
(B) 3
(C) 4
(D) 5
(E) 7


Ans: E

Solution: 47*49 = (48-1)*(48+1) = 48^2 -1^2
divisor = 8
48/8 = Reminder 0
1/8 reminder 1 put this in the equation = 0-1= -1..as remainder can not be negative so (8-1)= 7 is the remainder.
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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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New post 15 Jun 2015, 23:03
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pacifist85 wrote:
Hello,

Thank you for the explanations.

OK I can now understand what remainders for a divisor mean.
However, I still do not understand why when you divide -1^2 by 8 you get a remainder of 7. Isn't it the same as 1/8? This is the 0.125 and I cannot see how you end up with 7 as a remainder..


Here is a post on negative remainders that might interest you: http://www.veritasprep.com/blog/2014/03 ... -the-gmat/
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Re: What is the remainder when (47)(49) is divided by 8? [#permalink]

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Re: What is the remainder when (47)(49) is divided by 8?   [#permalink] 05 Dec 2017, 06:29
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