A number when divided successively by 4 and 5 leaves remainders 1 and

Question

A number when divided successively by 4 and 5 leaves remainders 1 and 4 respectively. What will be the remainder when this number is divided by 20?

(A) 0
(B) 3
(C) 4
(D) 9
(E) 17

v12345 · Accepted Answer

Here, the concept is of successive division
i.e. the no is first divided by 4 and it leaves remainder 1 and quotient is let x,

therefore we have  number = 4x + 1

and then the quotient x is divided by 5 and the remainder is 4

so, we have x in the form of  x = 5k + 4

putting this value of x above, we get

number = 4(5k + 4) +1

=> number = 20k + 17

when this number will be divided by 20, we will get  remainder = 17

Answer choice E

Kudos, if you like the explanation

JeffTargetTestPrep · Answer

We need to find a number that when divided by 4 leaves a remainder of 1 and when the quotient from this division is divided by 5 leaves a remainder of 4. Let’s represent this number by n. 
 
Since our number produces a remainder of 1 when divided by 4, it must be true that n = 4p + 1 for some integer p. 
 
Since the quotient from the previous division, which is p, produces a remainder of 4 when divided by 5, we have p = 5q + 4. Let’s substitute this expression of p into the previous equation:
 
n = 4p + 1 
 
n = 4(5q + 4) + 1 
 
n = 20q + 16 + 1
 
n = 20q + 17
 
Finally, since 20q is divisible by 20, the remainder from the division of n by 20 is 17.
 
Answer: E

BrentGMATPrepNow · Answer

When it comes to remainders, we have a nice rule that says:

If N divided by D, leaves remainder R, then the possible values of N are R, R+D, R+2D, R+3D,. . . etc.    
For example, if k divided by 5 leaves a remainder of 1, then the possible values of k are: 1, 1+5, 1+(2)(5), 1+(3)(5), 1+(4)(5), . . . etc.

A number when divided successively by 4 leaves a remainder 1 
Possible values of the number are: 1, 5,  9 , 13, 17, 21,...

A number when divided successively by 5 leaves a remainder 4 
Possible values of the number are: 4,  9 ...STOP!

Both lists contain  9 , so this could be the number.

What will be the remainder when this number is divided by 20? 
 9  divided by 20 = 0 with  remainder 9

Answer: D

Cheers,
Brent

Skywalker18 · Answer

I think the keyword here is that the number is divided  successively  .
Let x be the number which when successively divided by 4 and 5 gives 1 and 4 as remainder.

1. When we first divide the number by 4 .
 x = 4y +1
where y is quotient from the first division

2. Now as the division is done successively , we divide the quotient from the first step by 5
 y= 5z +4
where z is quotient from the second division
If Z=1 ,
y = 9
x= 9*4 + 1
 = 37

Now , on dividing the by 20 , we get 17

Answer E

Alternatively , we can also proceed from the final quotient .

Let x be the quotient after dividing the original number by 5.
So , the number which we divided by 5 to yield a remainder of 4 = 5x+ 4

This number 5x+4 is the quotient from the first division of the original number by 4
Therefore , the number should be = 4(5x+4) + 1
 = 20x +17

We can clearly see that when divided by 20 , we should get a remainder of 17

Answer E

BrentGMATPrepNow · Answer

I have to admit that, when I saw the word "successively," I thought that this might be what it means. However, I soon discounted that possibility and assumed the intent of the question was to tell us that, " When a certain number is divided by 4 and 5, the remainders are 1 and 4 respectively "

Here's why I discounted the other possibility:

In order for us to proceed as you have suggested above, we must ignore some important information. For example, let's divide 49 "successively" by 5 and 3. First, 49 divided by 5 equals 9 with remainder 4. This means that 49 divided by 5 equals 9 4/5. Now, we must take the result and divide by 3. You are suggesting that we take 9 and divide by 3. What about the the remainder of 4? What do we do with that? Alternatively, what do we do with the 4/5? We ignore it? What is there in the question that tells us to ignore the remainder from the first division?

We could also note that dividing a number successively by 5 and 3, is the same as dividing a number by 15. If we divide 49 by 15, we get a remainder of 4. If we apply the operations as you suggestabove, we get a remainder of 0.

The concept of successively dividing only seems to make sense (in my opinion) if the original number is divisible by BOTH divisors. For example, it makes sense to say that dividing 48 successively by 3 and 2 yields a result of 8. That is, 48/3 = 16 and 16/2 = 8.

Ignoring a certain component (i.e., the remainder) of the first division when performing the second division is not stated anywhere in the question. Given this, I don't think this could ever be an official GMAT question. I have certainly never seen a similar question. Have any other experts seen an official question of this nature?

Cheers,
Brent

zxcvbnmas · Answer

4x+1= 1,5,9,13,17,21,25 ,29 ,33,37,41,45, 49 ... 5x+4= 4,9,14,19,24, 29 ,39,44, 49 .... 29 & 49 are common in both so if we divide 29 or 49 by 20, we get 9 IMO Answer is D :roll:

Praveengeol · Answer

Using cross multiplication formula

4 5
 
1 4

(4*4)+1 = 17

Good to learn this formula from Bunuel Thread only. It works great.

Answer is E

adiagr · Answer

First it is important to understand successive division.

Given No. is divided by 4. Then whatever is the quotient, that quotient is divide by 5.

Let number is N.

N is of the type 4a+1 (Where a is any integer)

Now when N is divided by 4, we get "a" as the quotient.

so a is of the type: 5b+4

Ultimately N is of the type: 4*(5b+4) + 1

N = 20b+16+1 = 20b+17

when N is divided by 20, remainder will be 17.

E is the answer.

stonecold · Answer

Again ..! Arghhh..

One of those Veritas prep Out of bound Questions.
Personal Opinion =>Veritas prep questions are either 500 level or out of scope of the gmat.
How on earth are we supposed to solve this question.
The only way someone can solve this up on the test day this if he/she has a PHD in mathematics.
Sigh.!

Abhishek009  How do i solve this one?

Abhishek009 · Answer

Just posting my approach-

No algebra required for solving this question since all the options are less than 20...

Try each option individually -

(A) 0/4 = 0 (Quotient) & 0(Remainder) further 0/5 = 0 (Quotient) & 0(Remainder)

(B) 3/4 = 3 (Remainder)

(C) 4/4 = 1 (Quotient) & 0(Remainder)

(D) 9/4 = 2 (Quotient) & 1 (Remainder) further 2/5 = 2 (Quotient) & 1(Remainder)

(E) 17/4 = 4 (Quotient) & 1 (Remainder) further 4/5 = 4 (Remainder)

Thus 17/20 will have remainder as 17, hence answer will be (E)...

For an algebraic approach find  Here

anshul1989 · Answer

hi experts , Please let me know if this is the right way to solve :

N/4= a(1) ------ (we can write this 4a+1=N) let N be any integer, a be the quotient and 1 is the remainder we get when we divide it by 4, as per question
now, a/5=b(4) ----- now the quotient 'a' is divided by 5 giving remainder 4.(let 'b' be the quotient in this case)
So, a/5=b(4) ----> we can write this as 5b+4 = a (possible value of a is : 4,9,14...)

lets put these values of a in equation 4a+1 = N ---> we get  N = 17,37,57

All these values of N when divide by 20 ... gives us a remainder 17 hence answer is E.
Please explain if any shortcut is there to solve these kind of problem.

Fdambro294 · Answer

Dividing successively:

1st). Take the number N and divide it by 4 ——yield quotient of A and a remainder of 1

N/4 = a + 1/4

N = 4a + 1

2nd) then take the quotient of A and divide by 5 ———yield quotient of B and a remainder of 4

a/5 = b + 4/5

a = 5b + 4

Insert the value of a into the first equation

N = 4(5b + 4) + 1

N = 20b + 17

Lastly, divide N by 20———> N = (Multiple of 20) + 17

Remainder = 17

(E)

Posted from my mobile device

aditya khanna · Answer

You have missed the word successively , which change the method of solving

dhruva09 · Answer

Hi  Bunuel , are there any similar sums available ?

Bunuel · Answer

Remainders

Theory  
 
  REMAINDERS  
  Remainders: Tips and hints  
  Divisibility and Remainders on the GMAT  
  Remainders on the GMAT  
  A Remainders Shortcut for the GMAT  
  All About Negative Remainders on the GMAT  
  A Tricky Question on Negative Remainders  
  Compilation of tips and tricks to deal with remainders  
  Cyclicity and the remainders on the GMAT  
  GMAT Question Patterns: Remainders

Questions  
 
  PS Questions  
  Units digits, exponents, remainders problems

For other subjects:  
 
  ALL YOU NEED FOR QUANT ! ! !  
  Ultimate GMAT Quantitative Megathread

Hope it helps.

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