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# A number when divided successively by 4 and 5 leaves

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A number when divided successively by 4 and 5 leaves [#permalink]  28 Aug 2012, 21:32
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A number when divided successively by 4 and 5 leaves remainder 1 and 4 respectively. when it is successively divided by 5 and 4, then the respective remainders will be;

A. 1,2
B. 2,3
C. 3,2
D. 4,1
E. Data not sufficient
[Reveal] Spoiler: OA

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- Stne

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Re: A number when divided successively by 4 and 5 leaves [#permalink]  29 Aug 2012, 00:12
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stne wrote:
A number when divided successively by 4 and 5 leaves remainder 1 and 4 respectively. when it is successively divided by 5 and 4, then the respective remainders will be;

A. 1,2
B. 2,3
C. 3,2
D. 4,1
E. Data not sufficient

When dividing a positive integer $$n$$ by another positive integer $$D$$ (divider), we obtain a quotient $$Q$$, which is a non-negative integer and a remainder R, which is an integer such that $$0\leq{R}<D$$. We can write $$n=DQ+R.$$

When dividing our number $$n$$ by 4 we obtain a remainder of 1, so, if the quotient is some integer $$Q$$, we can write $$n=4Q+1.$$
Now, dividing $$Q$$ by 5, we obtain another quotient say $$q$$ and remainder 4, thus we can write $$Q=5q+4.$$

It follows that $$n=4(5q+4)+1=20q+17.$$
Since $$n=20q+17=5(4q+3)+2$$, it means that when dividing $$n$$ by 5 first, we get a quotient $$4q+3$$ and remainder 2.
Then dividing $$4q+3$$ by 4 we obviously obtain a remainder of 3.

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Re: A number when divided successively by 4 and 5 leaves [#permalink]  29 Aug 2012, 21:09
EvaJager wrote:
stne wrote:
A number when divided successively by 4 and 5 leaves remainder 1 and 4 respectively. when it is successively divided by 5 and 4, then the respective remainders will be;

A. 1,2
B. 2,3
C. 3,2
D. 4,1
E. Data not sufficient

When dividing a positive integer $$n$$ by another positive integer $$D$$ (divider), we obtain a quotient $$Q$$, which is a non-negative integer and a remainder R, which is an integer such that $$0\leq{R}<D$$. We can write $$n=DQ+R.$$

When dividing our number $$n$$ by 4 we obtain a remainder of 1, so, if the quotient is some integer $$Q$$, we can write $$n=4Q+1.$$
Now, dividing $$Q$$ by 5, we obtain another quotient say $$q$$ and remainder 4, thus we can write $$Q=5q+4.$$

It follows that $$n=4(5q+4)+1=20q+17.$$
Since $$n=20q+17=5(4q+3)+2$$, it means that when dividing $$n$$ by 5 first, we get a quotient $$4q+3$$ and remainder 2.
Then dividing $$4q+3$$ by 4 we obviously obtain a remainder of 3.

Thanks for the solution eva ..+1
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Re: A number when divided successively by 4 and 5 leaves [#permalink]  29 Aug 2012, 21:25
Expert's post
stne wrote:
A number when divided successively by 4 and 5 leaves remainder 1 and 4 respectively. when it is successively divided by 5 and 4, then the respective remainders will be;

A. 1,2
B. 2,3
C. 3,2
D. 4,1
E. Data not sufficient

You can solve it by checking for a number.

What do you mean by 'divided successively by 4 and 5'? It means, you divide the number by 4 and then you divide the quotient obtained (and not the original number) by 5.

When divided by 4, the number leaves remainder 1 so the number must be of the form 5 or 9 or 13 or 17 or 21 etc. For each of these numbers, remainder will be 1 and quotient will be 1 or 2 or 3 or 4 or 5 respectively. When the quotient obtained is divided by 5, it leaves remainder 4. We can easily pin point this number if the quotient obtained is 4 itself. The remainder will be 4 in that case. Hence 17 satisfies the given condition.

When you divide 17 by 5, you get 2 remainder and 3 quotient. When you divide 3 by 4, you get 3 remainder.

Note: GMAT PS questions do not have 'data not sufficient' option.
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Re: A number when divided successively by 4 and 5 leaves [#permalink]  29 Aug 2012, 23:41
VeritasPrepKarishma wrote:
stne wrote:
A number when divided successively by 4 and 5 leaves remainder 1 and 4 respectively. when it is successively divided by 5 and 4, then the respective remainders will be;

A. 1,2
B. 2,3
C. 3,2
D. 4,1
E. Data not sufficient

You can solve it by checking for a number.

What do you mean by 'divided successively by 4 and 5'? It means, you divide the number by 4 and then you divide the quotient obtained (and not the original number) by 5.

When divided by 4, the number leaves remainder 1 so the number must be of the form 5 or 9 or 13 or 17 or 21 etc. For each of these numbers, remainder will be 1 and quotient will be 1 or 2 or 3 or 4 or 5 respectively. When the quotient obtained is divided by 5, it leaves remainder 4. We can easily pin point this number if the quotient obtained is 4 itself. The remainder will be 4 in that case. Hence 17 satisfies the given condition.

When you divide 17 by 5, you get 2 remainder and 3 quotient. When you divide 3 by 4, you get 3 remainder.

Note: GMAT PS questions do not have 'data not sufficient' option.

Maybe, it would have been better to replace E by "Cannot be determined".
In fact, this question can be easily transformed into a DS question:

What is the value of positive integer n?

(1) When divided by 4, n leaves a reminder of 1.
(2) After dividing n by 4, the quotient is further divided by 5 leaving a remainder of 4.
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Re: A number when divided successively by 4 and 5 leaves [#permalink]  31 Aug 2012, 15:25
EvaJager wrote:
VeritasPrepKarishma wrote:
stne wrote:
A number when divided successively by 4 and 5 leaves remainder 1 and 4 respectively. when it is successively divided by 5 and 4, then the respective remainders will be;

A. 1,2
B. 2,3
C. 3,2
D. 4,1
E. Data not sufficient

You can solve it by checking for a number.

What do you mean by 'divided successively by 4 and 5'? It means, you divide the number by 4 and then you divide the quotient obtained (and not the original number) by 5.

When divided by 4, the number leaves remainder 1 so the number must be of the form 5 or 9 or 13 or 17 or 21 etc. For each of these numbers, remainder will be 1 and quotient will be 1 or 2 or 3 or 4 or 5 respectively. When the quotient obtained is divided by 5, it leaves remainder 4. We can easily pin point this number if the quotient obtained is 4 itself. The remainder will be 4 in that case. Hence 17 satisfies the given condition.

When you divide 17 by 5, you get 2 remainder and 3 quotient. When you divide 3 by 4, you get 3 remainder.

Note: GMAT PS questions do not have 'data not sufficient' option.

Maybe, it would have been better to replace E by "Cannot be determined".
In fact, this question can be easily transformed into a DS question:

What is the value of positive integer n?

(1) When divided by 4, n leaves a reminder of 1.
(2) After dividing n by 4, the quotient is further divided by 5 leaving a remainder of 4.

Thank you eva and karishma

here is a matrix way which I found here, no need to search for a number by trial and error
directly obtain the number that is being divided (successively). By this method.

(i)arrange both the divisors in ascending order , and their respective remainders below it
a b
c d

(ii)next cross multiply diagonally like this d * a
(iii)and then add c this will give the number that is being divided ,( dividend )

so coming back to the given question we have a=4 , b = 5 , c= 1 and d = 4

4 5
1 4

so 4*4 +1 = 17 the original number we are dividing ( dividend )

17/4 = quotient 4 and remainder 1 ( as stated )
quotient 4 divided by 5 remainder 4 ( as stated )

so this must be the number as it meets both the conditions , now that we know the number it becomes relatively easy.

Now the question becomes when 17 is divided successively by 5 and 4 what are the remainders ?

now 17/5 = quotient 3 remainder 2

quotient 3 divided by 4 remainder 3 , hence when 17 is divided successively by 5 and 4 , the remainders are 2 and 3 , and that is required answer.
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Re: A number when divided successively by 4 and 5 leaves [#permalink]  02 Sep 2012, 21:18
Expert's post
stne wrote:
Thank you eva and karishma

here is a matrix way which I found here, no need to search for a number by trial and error
directly obtain the number that is being divided (successively). By this method.

(i)arrange both the divisors in ascending order , and their respective remainders below it
a b
c d

(ii)next cross multiply diagonally like this d * a
(iii)and then add c this will give the number that is being divided ,( dividend )

so coming back to the given question we have a=4 , b = 5 , c= 1 and d = 4

4 5
1 4

so 4*4 +1 = 17 the original number we are dividing ( dividend )

17/4 = quotient 4 and remainder 1 ( as stated )
quotient 4 divided by 5 remainder 4 ( as stated )

so this must be the number as it meets both the conditions , now that we know the number it becomes relatively easy.

Now the question becomes when 17 is divided successively by 5 and 4 what are the remainders ?

now 17/5 = quotient 3 remainder 2

quotient 3 divided by 4 remainder 3 , hence when 17 is divided successively by 5 and 4 , the remainders are 2 and 3 , and that is required answer.

The 'matrix method' is exactly the same as the other methods. It is just a different representation. There is nothing wrong with it but can you really 'learn' a different method for every type of question you come across? If you can, go ahead. There is absolutely nothing wrong with it.

The matrix method is finding the first number.

If we put n in the equation form using a, b, c and d (as used in the matrix method by you), we get n = a(bm + d) + c = 4(5m + 4) + 1 [as shown by EvaJager above]
Here, to get the first value of n, you are just assuming m = 0. You get n = ad + c = 4*4 + 1 (this is what your matrix method does)
I also did the same to get the value of 17. I assumed that the quotient when you divide by 5 is 0 and all you have to worry about is the remainder of 4. So you just have to obtain the quotient as 4 in the previous step.

As I said, the method is no different and there is nothing wrong with it. If you do want to use it, make sure you understand why you are multiplying a and d together and then adding c.
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Re: A number when divided successively by 4 and 5 leaves [#permalink]  17 Oct 2012, 02:42
Can the matrix method suggested by stne also be used in the case where we are dividing the number three times?

For example:

On dividing a certain number by 5,7,8 successively, the remainders are 2,3 and 4 respectively. What would be the remainders if order of division is reversed.
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Re: A number when divided successively by 4 and 5 leaves [#permalink]  17 Oct 2012, 03:34
karishmasparmar wrote:
Can the matrix method suggested by stne also be used in the case where we are dividing the number three times?

For example:

On dividing a certain number by 5,7,8 successively, the remainders are 2,3 and 4 respectively. What would be the remainders if order of division is reversed.

(1) I don't think such a question can appear on the real GMAT test.
(2) I don't know how the method proposed by stne would work in this case. Even the algebra can get quite messy...
$$n=5(7(8N+4)+3)+2=5\cdot{7}\cdot{8}N+157$$, for some positive integer $$N$$.
In reverse order, when dividing by 8,7,5 successively, first quotient $$5\cdot{7}N+19$$ and remainder 5.
Then, dividing by 7, quotient $$5N+3$$ and remainder 4.
Finally, when dividing by 5, quotient $$N$$ and remainder 3.
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Re: A number when divided successively by 4 and 5 leaves [#permalink]  17 Oct 2012, 20:28
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KUDOS
Expert's post
karishmasparmar wrote:
Can the matrix method suggested by stne also be used in the case where we are dividing the number three times?

For example:

On dividing a certain number by 5,7,8 successively, the remainders are 2,3 and 4 respectively. What would be the remainders if order of division is reversed.

Most certainly! As I said in my post before, the matrix method is just another representation of algebra/logic. Finally you are doing the same thing in every case.

Divisors: a, b, c
Remainders: m, n, p

a b c
m n p

Use matrix method twice: First focus on two right columns: you get b*p + n. Then multiply this result i.e. (b*p + n) by a and add m to it to get a*(b*p + n) + m

5 7 8
2 3 4

First step: 7*4 + 3 = 31
Second step 5*31 + 2 = 157

157 is the first such number. Divide 157 by 8, 7, 5 in that order:
157/8 Quotient = 19, Remainder = 5
19/7 Quotient = 2, Remainder = 5
2/5 Quotient = 0, Remainder = 2

Remainders are 5, 5, 2

But preferably, don't use this method. Just use logic. Why? Because, say, you knew this 'matrix method' and used it to solve 2 divisor problems. What if you get 3 divisor problems in the test? Can you use matrix method at that time? No! I derived the method for 3 divisor problems by first working out the logic and then putting it in the matrix form. So why not use logic only?

Think this way:

Divisors: 5, 7, 8
Remainders: 2, 3, 4

You start by considering the last step first. At the end, when you divide by 8, you want remainder to be 4 (let's assume the quotient we get is 0 in this case). This means that after division by 7, we should get 4 as the quotient (so that we can get 4 as remainder when we divide by 8 in the next step). So, when you divide by 7, the quotient must be 4 and remainder must be 3. At this step, the number must be 7*4 + 3 = 31
Now, when you divide by 5, you need the quotient to be 31 and remainder to be 2 so number must be 5*31 + 2 = 157
157 is the first such number. (If this is not very clear, divide 157 successively by 5, 7, and 8 and see what goes on).

Now you proceed as before!

I hope you see that you can follow the same logic and solve even with 5 or more divisors very quickly!

Also, there is nothing wrong with 'such a question'. It's very logical and hence fair game as far as GMAT is concerned.
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Re: A number when divided successively by 4 and 5 leaves   [#permalink] 17 Oct 2012, 20:28
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