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Math Expert V
Joined: 02 Sep 2009
Posts: 60460
What is the remainder when 1555 * 1557 * 1559 is divided by 13?  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 63% (02:06) correct 37% (02:06) wrong based on 294 sessions

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What is the remainder when 1555 * 1557 * 1559 is divided by 13?

(A) 0
(B) 2
(C) 4
(D) 9
(E) 11

Kudos for a correct solution.

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Math Expert V
Joined: 02 Sep 2009
Posts: 60460
Re: What is the remainder when 1555 * 1557 * 1559 is divided by 13?  [#permalink]

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Bunuel wrote:
What is the remainder when 1555 * 1557 * 1559 is divided by 13?

(A) 0
(B) 2
(C) 4
(D) 9
(E) 11

Kudos for a correct solution.

VERITAS PREP OFFICIAL SOLUTION

Since it is a GMAT question (a question for which we will have no calculator), multiplying the 3 numbers and then dividing by 13 is absolutely out of question! There has to be another method.

Say n = 1555 * 1557 * 1559

When we divide 1555 by 13, we get a quotient of 119 (irrelevant to our question) and remainder of 8. So the remainder when we divide 1557 by 13 will be 8+2 = 10 (since 1557 is 2 more than 1555) and when we divide 1559 by 13, the remainder will be 10+2 = 12 (since 1559 is 2 more than 1557).

So n = (13*119 + 8)*(13*119 + 10)*(13*119 + 12) (you can choose to ignore the quotient and just write it as ‘a’ since it is irrelevant to our discussion)

So we need to find the remainder when n is divided by 13.

Note that when we multiply these factors, all terms we obtain will have 13 in them except the last term which is obtained by multiplying the constants together i.e. 8*10*12.

Since all other terms are multiples of 13, we can say that n is 8*10*12 (= 960) more than a multiple of 13. There are many more groups of 13 balls that we can form out of 960.

960 divided by 13 gives a remainder of 11.

Hence n is actually 11 more than a multiple of 13.

We did not use the negative remainders concept here. Let’s see how using negative remainders makes our calculations easier here. The remainder of 8, 10 and 12 imply that the negative remainders are -5, -3 and -1 respectively.

Now n = (13a – 5) * (13a – 3) * (13a – 1)

The last term in this case is -5*-3*-1 = -15

This means that n is 15 less than a multiple of 13 i.e. actually 2 less than a multiple of 13 because when you go back 13 steps, you get another multiple of 13. This gives us a negative remainder of -2 which means the positive remainder in this case will be 11.
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Re: What is the remainder when 1555 * 1557 * 1559 is divided by 13?  [#permalink]

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Bunuel wrote:
What is the remainder when 1555 * 1557 * 1559 is divided by 13?

(A) 0
(B) 2
(C) 4
(D) 9
(E) 11

Kudos for a correct solution.

1555/13--->Remainder=8
1557/13--->Remainder=10
1559/13--->Remainder=12

8*10*12=960/13--->Remainder=11

##### General Discussion
Manager  Joined: 15 May 2014
Posts: 61
What is the remainder when 1555 * 1557 * 1559 is divided by 13?  [#permalink]

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$$\frac {1555\,*\,1557\,*\,1559}{13} = \frac {(1560-5) *(1560-3)*(1560-1)}{13}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$=$$ $$(multiples\,of\,13$$) + $$(-5*-3*-1)$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$=$$ $$(multiples\,of\,13$$) - $$15$$

here the last term $$-15$$ is a negative number
$$\frac{-15}{13}\,=\,quotient\,*13\,+\,remainder$$
here remainder should be $$0\,\leq\,remainder\,<\,13$$
So $$-15\,=\,13\,(-2)\,+\,11$$
Remainder $$= 11$$

Originally posted by sudh on 18 Apr 2015, 03:18.
Last edited by sudh on 18 Apr 2015, 21:07, edited 3 times in total.
Manager  Joined: 27 Dec 2013
Posts: 193
Re: What is the remainder when 1555 * 1557 * 1559 is divided by 13?  [#permalink]

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1
I liked the way you solved the question. I did the same except that I elaborated the solution and ended up with some big numbers +960.

Everyday is learning . Nice work.

AmoyV wrote:
Bunuel wrote:
What is the remainder when 1555 * 1557 * 1559 is divided by 13?

(A) 0
(B) 2
(C) 4
(D) 9
(E) 11

Kudos for a correct solution.

1555/13--->Remainder=8
1557/13--->Remainder=10
1559/13--->Remainder=12

8*10*12=960/13--->Remainder=11

Manager  Joined: 17 Mar 2015
Posts: 113
What is the remainder when 1555 * 1557 * 1559 is divided by 13?  [#permalink]

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sudh wrote:
$$\frac {1555\,*\,1557\,*\,1559}{13} = \frac {(1560-5) *(1560-3)*(1560-1)}{13}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$=$$ $$(multiples\,of\,13$$) + $$(-5*-3*-1)$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$=$$ $$(multiples\,of\,13$$) - $$15$$

here the last term $$-15$$ is a negative number
So
$$-13 \,\geq remainder \,\leq -26$$
Since remainder should be a positive less than divisor
$$-26\,+\,11\,=\,-15$$
Remainder $$= 11$$

I am abit confused with the way you found an answer, maybe I'm not good at remainders but imo you can transform your expression like this to make it "easier", I guess:
$$(multiples\,of\,13$$) - $$15$$ = $$(multiples\,of\,13$$) - $$13$$ - $$2$$ = $$(multiples\,of\,13$$) - $$13$$ - $$13$$ + $$11$$, which lets us explicitly figure out that the ending result of division is $$"integer" - 2 + 11/13$$ which pretty much tells us that the remainder is 11.
Ty for the solution though, pretty neat.
Manager  Joined: 15 May 2014
Posts: 61
What is the remainder when 1555 * 1557 * 1559 is divided by 13?  [#permalink]

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Zhenek wrote:
sudh wrote:
$$\frac {1555\,*\,1557\,*\,1559}{13} = \frac {(1560-5) *(1560-3)*(1560-1)}{13}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$=$$ $$(multiples\,of\,13$$) + $$(-5*-3*-1)$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$=$$ $$(multiples\,of\,13$$) - $$15$$

here the last term $$-15$$ is a negative number
So
$$-13 \,\leq remainder \,\leq -26$$
Since remainder should be a positive less than divisor
$$-26\,+\,11\,=\,-15$$
Remainder $$= 11$$

I am abit confused with the way you found an answer, maybe I'm not good at remainders but imo you can transform your expression like this to make it "easier", I guess:
$$(multiples\,of\,13$$) - $$15$$ = $$(multiples\,of\,13$$) - $$13$$ - $$2$$ = $$(multiples\,of\,13$$) - $$13$$ - $$13$$ + $$11$$, which lets us explicitly figure out that the ending result of division is $$"integer" - 2 + 11/13$$ which pretty much tells us that the remainder is 11.
Ty for the solution though, pretty neat.

Sorry for the confusion

$$\frac{-15}{13}\,=\,quotient\,*13\,+remainder$$
here remainder should be $$0\,\leq\,remainder\,<\,13$$
So $$-15\,=\,13\,(-2)\,+\,11$$
Or we could just borrow $$(2*13)\,=\,26$$ from the $$(multiples\,of\,13$$) and add them with $$-15$$, giving us the remainder $$11$$
Intern  Joined: 18 Aug 2015
Posts: 6
Re: What is the remainder when 1555 * 1557 * 1559 is divided by 13?  [#permalink]

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Find out the remainders for individual terms: 8*10*12/13 = 80*12/13 = 2*12/13 = 24/13 = 11
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Re: What is the remainder when 1555 * 1557 * 1559 is divided by 13?  [#permalink]

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Bunuel wrote:
What is the remainder when 1555 * 1557 * 1559 is divided by 13?

(A) 0
(B) 2
(C) 4
(D) 9
(E) 11

Kudos for a correct solution.

i tried to figure out which option is the fastest...
then came up with this one...
13 => 1300 is divisible by 13
1430 is divisible by 3
1560 is divisible by 3.

1555 = 1560-5, which means, if divided by 13, we'll have a remainder of 8
1557 = 1560-3, meaning that if divided by 13, we'll have a remainder of 10
1559 = 1560-1, meaning that if divided by 13, we'll have a remainder of 12.
now..8*10*12 = or 80*(10+2) = 800+160=960.
960/13 = 73, with a remainder of 11.
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Re: What is the remainder when 1555 * 1557 * 1559 is divided by 13?  [#permalink]

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Bunuel wrote:
What is the remainder when 1555 * 1557 * 1559 is divided by 13?

(A) 0
(B) 2
(C) 4
(D) 9
(E) 11

Kudos for a correct solution.

$$\frac{1555}{13}$$ = Remainder $$8$$
$$\frac{1557}{13}$$ = Remainder $$10$$
$$\frac{1559}{13}$$ = Remainder $$12$$

Finally we have $$\frac{8*10*12}{13}$$ = Remainder 11, Answer will be (E)

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Re: What is the remainder when 1555 * 1557 * 1559 is divided by 13?  [#permalink]

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Bunuel wrote:
What is the remainder when 1555 * 1557 * 1559 is divided by 13?

(A) 0
(B) 2
(C) 4
(D) 9
(E) 11

Kudos for a correct solution.

Solved it this way -

Divide each of the number by 13.. You will get 8,10,12 as remainder. Now, multiply the remainders and again Divide by 13.
You will get 11 as remainder.
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Re: What is the remainder when 1555 * 1557 * 1559 is divided by 13?  [#permalink]

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Bunuel wrote:
What is the remainder when 1555 * 1557 * 1559 is divided by 13?

(A) 0
(B) 2
(C) 4
(D) 9
(E) 11

Kudos for a correct solution.

R(1555 * 1557 * 1559/3)

Remainder when 1555 is divided by 13 = 8
Remainder when 1557 is divided by 13 = 10 (or -3)
Remainder when 1559 is divided by 13 = 12 (or -1)

Remainder [8*(-3)*(-1)/13] = R (24/13) = 11

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Re: What is the remainder when 1555 * 1557 * 1559 is divided by 13?  [#permalink]

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Bunuel wrote:
What is the remainder when 1555 * 1557 * 1559 is divided by 13?

(A) 0
(B) 2
(C) 4
(D) 9
(E) 11

We don’t have to multiply the numbers and then divide the product by 13. We can divide each factor by 13 first.

Notice that when 1555 is divided by 13, the remainder is 8 (with quotient = 119). Thus, the remainders are 10 and 12 when 1557 and 1559 are divided by 13, respectively. Now we multiply these remainders and divide the product by 13.

Since 8 x 10 x 12 = 960 and when 960 is divided by 13, the remainder is 11 (with quotient = 73). Thus, the remainder, when 1555 x 1557 x 1559 is divided by 13, must also be 11.

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