On dividing a certain number by 5, 7 and 8 successively, the remainder

Question

On dividing a certain number by 5, 7 and 8 successively, the remainders obtained are 2, 3 and 4 respectively

Third divisor = 8
Third remainder = 4
i.e. Number of this type = 8+4 = 12

Second divisor = 7
Quotient then = 12
Second remainder = 3
i.e. Number of this type = 12*7+3 = 87

First divisor = 5
Quotient then = 87
Second remainder = 2
i.e. Number of this type = 87*5+2 = 437

(A) 3, 3, 2
(B) 3, 4, 2
(C) 5, 4, 3
(D) 5, 5, 2
(E) 6, 4, 3

v12345 · Accepted Answer

Here, the concept is of successive division
i.e. the no is first divided by 5 and it leaves remainder 2 and quotient is let x,

therefore we have  number = 5x + 2  ---------------------- (1)

and then the quotient x is divided by 7 and the remainder is 3

so, we have x in the form of  x = 7y + 3

and then the quotient y is divided by 8 and the remainder is 4

so, we have x in the form of  y = 8z + 4

putting this value of x and y in (1) above, we get

number = 5(7(8z +4) + 3) +2 
 => number = 5(56 z +31) +2 
 => number = 280 z + 157

when this number will be  divided by 8, we will get remainder = 5  and quotient = 35 z + 19

when this quotient will be  divided by 7, we will get remainder = 5  and quotient = 5 z + 2

when this quotient will be  divided by 5, we will get remainder = 2

Answer choice D

Kudos, if you like the explanation

Praveengeol · Answer

Cross Multiplication Formula

5 7 8
2 3 4

(4*7)+3=31
(31*5)+2 = 157

Number is 157

Successive division of 157 by 8,7 and 5 will give remainder 5,5 and 2.

ANSWER IS D.

Skywalker18 · Answer

The number which we divided by 8 will be quotient from division by 7 = 8x + 4
The number which we divided by 7 will be quotient from division by 5 = 7(8x+4) + 3 = 56x +31
The original number which we divided by 5 =5(56x +31) + 2 
 = 280x +155 +2
 = 280x + 157

When this number is divided by 8 = ( 280x+ 157)/8 = 35x + 19 and a remainder of 5
On dividing by 7 , ( 35x + 19) / 7 = 5x + 2 and a remainder of 5
On Dividing by 5 , (5x+2)/5 = x and a remainder of 2

Therefore the remainders will be 5,5 and 2

Answer D

GMATinsight · Answer

On dividing a certain number by 5, 7 and 8 successively, the remainders obtained are 2, 3 and 4 respectively

Third divisor = 8
Third remainder = 4
i.e. Number of this type = 8+4 = 12

Second divisor = 7
Quotient then = 12
Second remainder = 3
i.e. Number of this type = 12*7+3 = 87

First divisor = 5
Quotient then = 87
Second remainder = 2
i.e. Number of this type = 87*5+2 = 437

Now, successively divide by 8, 7 and 5

837 divided by 8 leaves Quotient 54 and remainder=5

54 divided by 7 leaves Quotient 7 and remainder=5

7 divided by 5 leaves Quotient 1 and remainder=2

Successive Remainders (5, 5, 2)
 
Answer: option D

MathRevolution · Answer

Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer.

On dividing a certain number by 5, 7 and 8 successively, the remainders obtained are 2, 3 and 4 respectively

Third divisor = 8
Third remainder = 4
i.e. Number of this type = 8+4 = 12

Second divisor = 7
Quotient then = 12
Second remainder = 3
i.e. Number of this type = 12*7+3 = 87

First divisor = 5
Quotient then = 87
Second remainder = 2
i.e. Number of this type = 87*5+2 = 437

(A) 3, 3, 2
(B) 3, 4, 2
(C) 5, 4, 3
(D) 5, 5, 2
(E) 6, 4, 3

Let the original number be X. Then by the successive dividing we have followings :
X=5A+2
A=7B+3
B=8C+4.

So we have A=7*(8C+4)+3=7*8C+31, and X=5*(7*8C + 31)+2 = 5*7*8C + 157.

Now by dividing X by 8, 7, 5 successively we have followings : 
X=8*(5*7C+19)+5
5*7C+19=7*(5C+2) + 5
5C+2=5C+2.

The remainders are, therefore, 5, 5, 2. 
The answer is (D).

OR

As dividing X by 8, 7, 5, we can ignore 5*7*8C(since 5*7*8C is a common multiple of 8, 7, 5, 5*7*8C doesn’t affect the remainders ). 
So we can consider only 157. 
157=8*19 + 5
19=7*2 + 5
2=5*0+2.
The remainders are 5, 5, 2.

mvictor · Answer

this is the first time when I see such a question...
my approach:
X is the number
X=5Q+2
5Q+2 = 7Z +3
7Z+3 = 8M + 4

so X=8M+4

where did I go wrong?
I believe the question is not worded properly: is the quotient only divided or together with the reminder?

gauravsaggis1 · Answer

mvictor , according to your approach you are dividing the same no. by 8,7 and 5 not successively.i.e.
x=5q+2,
5q+2=7z+3=x=8m+4
for dividing successively we need to divide quotient *multiple of the previous no. as explained by mathrevolution and skywalker18.

sudhirmadaan · Answer

to find a number first start with division by 8
when number is divided by 8 remainder is 4, so number is = 4, which is Q quotient 
when number which is divided by 7 and leaver a remainder 3 would be 7*4 +3 = 31 , which quotient for the number divided by 5
when number which is divided by 5 and leaver a remainder 2 would be 5*31 +2 = 157

Now 157 / 8 remainder is 5 and Q quotient is 19 , next is 19/7 remainder is 5
D is the answer.

KarishmaB · Answer

For a discussion on the concept of successive division and this question, check: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2012/10 ... -division/

adiagr · Answer

In successive division, the quotient will become the next dividend.

Suppose number is N

Then N = 5x+2; Next dividend x.

x = 7y+3; Next dividend y

y = 8z+4

so N = 5* {7*(8z+4) +3} +2 = 5 * (56z+31) +2 = 280 z +157

When N is divided by 8, the quotient is: 8*(35z+19) +   5

First Remainder is 5.

So options A, B E are out.

Quotient is 35z+19; This becomes next dividend.

Divide it by 7: 7 *(5z+2) +   5

Remainder is 5. --> So option C is out

D is the answer.

chandan1988 · Answer

Bunuel  what is the significance of the line "When the order of division is reversed" in the question?

Bunuel · Answer

First we divide x by 5, 7 and 8 successively, and getting the remainders of 2, 3 and 4 respectively.
Then we are dividing x by 8, 7, and 5 successively(so we are dividing by the same divisors but in reverse order), and getting the remainders of 8, 7 and 5, respectively.

For more on successive dividing check  Successive Division  post.

Hope it helps.

Andrewcoleman · Answer

Hi  VeritasKarishma ,  Bunuel !

I got this right! and I understand the set up and all of the logic perfectly, but it took me ~3 to 4 minutes to get.

Do you recommend anything to get this time down?

Thanks!!!

KarishmaB · Answer

On dividing a certain number by 5, 7 and 8 successively, the remainders obtained are 2, 3 and 4 respectively.

N = 5a + 2 = 5(7b + 3) + 2 = 5(7(8c + 4) + 3) + 2

To get first such number, put c = 0.
First such number = 157

157/8 gives quotient 19 and leaves remainder 5 (between options (C) and (D))
19/7 leaves remainder 5

Answer (D)

Shouldn't take more than 2 mins.

Andrewcoleman · Answer

thanks!

I practiced it a few times and got it no problem!

Great post!

ScottTargetTestPrep · Answer

Solution:

We need to find a number that when it is divided by 5, 7 and 8 successively, the remainders obtained are 2, 3 and 4 respectively.

We have to go backward. That is, first we have to find a positive number that, when divided by 8, yields a remainder of 4. We see that the smallest such number is 4. Next, we need to find a number that, when divided by 7, yields a quotient of 4 and a remainder of 3. We see that number is 7 x 4 + 3 = 31. Finally, we need to find a number that, when divided by 5, yields a quotient of 31 and a remainder of 2. We see that number is 5 x 31 + 2 = 157.

Since we’ve found 157 using a “backward” process. We can check whether it actually works:

157/5 = 31 R 2

31/7 = 4 R 3

4/8 = 0 R 4

So 157 actually works. Now we can divide it by 8, 7, and 5, successively, and find the respective remainders:

157/8 = 19 R 5

19/7 = 2 R 5

2/5 = 0 R 2

Alternate Solution:
 
Let x be the number. Since x yields a remainder of 2 when divided by 5, we can write x = 5k + 2 for some positive integer k.

Since k yields a remainder of 3 when divided by 7, we can write k = 7s + 3 for some positive integer s.

Since s yields a remainder of 4 when divided by 8, we can write s = 8m + 4 for some positive integer m.

Let’s substitute s = 8m + 4 in k = 7s + 3:

k = 7(8m + 4) + 3

k = 56m + 28 + 3

k = 56m + 31

Let’s substitute k = 56m + 31 in x = 5k + 2:

x = 5(56m + 31) + 2

x = 280m + 155 + 2

x = 280m + 157

Now, let’s find the remainder when x is divided by 8. Let’s write:

x = 8 * 35m + 152 + 5

x = 8 * 35m + 8 * 19 + 5

x = 8(35m + 19) + 5

So, the remainder when x is divided by 8 is 5. We eliminate answer choices A, B and E.

Next, let’s divide the quotient from division by 8 (which is 35m + 19) by 7:

35m + 19 = 7* 5m + 14 + 5

35m + 19 = 7(5m + 2) + 5

So, the remainder when 35m + 19 is divided by 7 is also 5. We eliminate answer choice C and conclude that the correct answer is D.

Answer: D

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On dividing a certain number by 5, 7 and 8 successively, the remainder

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