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# What is the remainder when 47*49 is divided by 8 ?

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Re: What is the remainder when 47*49 is divided by 8 ? [#permalink]
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mun23 wrote:
What is the remainder when (47)(49) is divided by 8?
(A)1
(B)3
(C)4
(D)5
(E)7

I have an alternate learnt here

$$Rof (47) (49)$$ when divided by 8

Always reminder of product of two numbers divided by a divisor will be equal to product of reminders of corresponding numbers divided by same divisor

Using this rule,
$$Rof (47)$$ = 7
$$Rof (49)$$ = 1

So$$Rof (47) (49)$$ = 7*1 =7

Other way would be
$$Rof (47)$$ = -1 ( which is same as 7, but to do simpler we use -ve number 8-7)
$$Rof (49)$$= 1
So$$Rof (47) (49)$$ = -1*1 =-1
Therefor 8-1= 7

I'm missed the link from which i learnt this..
I'm searching for it.. Once i get it will post it up here !!
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Re: What is the remainder when 47*49 is divided by 8 ? [#permalink]
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Let N = A * B and we would like to find the remainder when N is divided by D.
A = aD + x where x is the remainder of A/D.
B = bD + y where y is the remainder of B/D.

so N = (aD + x) * (bD + y)
= abD$$^2$$ + bxD + ayD + xy

So, N/D will give a remainder xy if xy < D.

Considering the above principle,

R1 = 47/8 = 7
R2 = 49/8 = 1
R = R1 * R2 = 7 * 1 = 7.
Hence option E will be the correct answer.

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

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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

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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

Kudos for a correct solution.
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Re: What is the remainder when 47*49 is divided by 8 ? [#permalink]
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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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]
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"
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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:

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]
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.

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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..

Here is a post on negative remainders that might interest you: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2014/03 ... -the-gmat/
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Re: What is the remainder when 47*49 is divided by 8 ? [#permalink]
mun23 wrote:
What is the remainder when (47)(49) is divided by 8?
(A)1
(B)3
(C)4
(D)5
(E)7

$$\frac{47}{8}$$ = Remainder $$7$$
$$\frac{49}{8}$$ = Remainder $$9$$

$$\frac{63}{8}$$ = Remainder $$7$$

Thus, answer will be (E) Remainder $$7$$
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Re: What is the remainder when 47*49 is divided by 8 ? [#permalink]
mun23 wrote:
What is the remainder when (47)(49) is divided by 8?
(A)1
(B)3
(C)4
(D)5
(E)7

To start, we can separately divide 47 by 8 and 49 by 8.

47/8 = 5 remainder 7

49/8 = 6 remainder 1

To determine the remainder of the product of 47 and 49, we can multiply the remainders together and we have 7 x 1 = 7.

Alternate solution:

Notice that (47)(49) = (48 - 1)(48 + 1) = 48^2 - 1^2 = 48^2 - 1. Now notice that the first term, 48^2, is divisible by 8 since 48 is divisible by 8. Thus, the remainder will be the second term, -1. If a number has a remainder of -1 when it’s divided by 8, the remainder is actually -1 + 8 = 7.

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Re: What is the remainder when 47*49 is divided by 8 ? [#permalink]
mun23 wrote:
What is the remainder when (47)(49) is divided by 8?
(A)1
(B)3
(C)4
(D)5
(E)7

product of units digits=63
63/8 leaves remainder of 7
E
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Re: What is the remainder when 47*49 is divided by 8 ? [#permalink]
Alternatively, we can use the unit digit of 47 and 49

7*9 = 63

63/8 = 7 remainder 7.

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

47/8 remainder = 7
49/8 remainder = 1

remainders = 7 * 1