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

Question

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

AmoyV · Accepted Answer

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

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

Answer: E

Bunuel · Answer

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.

sudh · Answer

\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

Answer E

shriramvelamuri · Answer

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.

Reached the same place and had the same answer 11.

Everyday is learning . Nice work.

Zhenek · Answer

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.

sudh · Answer

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

RishiCTG · Answer

Find out the remainders for individual terms: 8*10*12/13 = 80*12/13 = 2*12/13 = 24/13 = 11

mvictor · Answer

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.
answer is E.

Abhishek009 · Answer

\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)

rahul16singh28 · Answer

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.

GMATinsight · Answer

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   = R (24/13) = 11

Answer: Option E

JeffTargetTestPrep · Answer

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.

Answer: E

MathRevolution · Answer

The question is based on the Divisibility and remainder theory concept. Also, as the numbers are bigger, one should know the divisibility application.

Divisibility rule of 13: Remove the last digit from a number, multiply it by 4, add the product to the truncated original number and continue this process until two digits remain. If the result is divisible by 13, then the original number is divisible by 13.

1555 => 155+ (5*4) = 175 => 17+(5*4)= 37 . ( On dividing by 13, remainder is either 11 or (-2).
1557 => 155+ (7*4) = 183 => 18+(3*4)= 30 . ( On dividing by 13, remainder is either 4 or (-9).
1559 => 155+ (9*4) = 191 => 19+(1*4)= 23 . ( On dividing by 13, remainder is either 10 or (-3).

Remainder :  \frac{(-2)*(-9)*(-3) }{ 13}  OR Remainder :  \frac{(11)*(4)*(10) }{ 13}

=>  \frac{(-54) }{ 13}  OR =>  \frac{(440) }{ 13}

=>  \frac{(-2) }{ 13}  OR =>  \frac{(33*13+11) }{ 13}

Remainder (-2) + 13 = 11 OR Remainder = 11

Answer E

GMATGuruNY · Answer

What is the remainder when 30 is divided by 4?

One approach:
1. Break the dividend 30 into factors: 30 = 5*6
2. Divide the divisor 4 into each factor:  \frac{5}{4}  = 1 R1,  \frac{6}{4}  = 1 R2
3. Multiple the resulting remainders: 1*2 = 2

Step 3 indicates that 30 divided by 4 will yield a remainder of 2.
This approach can be applied to any problem that asks for the remainder when a large integer is divided by a divisor.
Repeat the 3 steps until the value yielded by Step 3 is less than the divisor.

\frac{1555}{13}  = 119  R8 
Since 1557 is 2 more than 1555, dividing by 13 will increase the remainder by 2:  R10 
Since 1559 is 2 more than 1557, dividing by 13 will increase the remainder again by 2:  R12

Multiplying the remainders in blue, we get:
8*10*12 = 960

Dividing 13 into 960, we get:
 \frac{960}{13}  = 73  R11

The value in green is less than the divisor (13) and thus is the desired remainder.

E

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

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

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