A Number When Divided Successively by 4 and 5 Leaves Remainder 1 and 4 Respectively. GMAT Problem Solving

Question: 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;

1. 1,2
2. 2,3
3. 3,2
4. 4,1
5. Data not sufficient

Correct Answer: B

Solution and Explanation
Approach Solution 1:

To answer this kind of question, put the Questions words into mathematical forms, then go through the solution step-by-step to get the remainder. Dividend = Divisor x Quotient + Remainder

Let the number be X.
When you divide X by 4 it gives remainder 1.
Let the divisor here be Y,
X = 4Y + 1 ….
Now, we obtain the remaining 4 when we divide Y by 5.
We consider the quotient to be 1, since 5 is evenly divided by Y.
Y = 1 × 5 + 4 = 9
Putting Y in (1) we get the equation as follows,
X = 4 × 9 + 1 = 37
When the number 37 is split by 5 and 4, the remaining number is
37 = 5 × 7 + 2. . . . . . . . . . . . . . . (a)
7 = 4 × 1 + 3. . . . . . . . . . . . . . . . (b)
We may infer from the above that option B is the correct one.
From (a) and (b), we can infer that when 37 is divided by 5, the result is 7, and when 7 is divided by 4, the result is 3. Because the numbers in the question are simply swapped, caution is advised when solving this type of problem.

Approach Solution 2:

Let me give you another method:

write the numbers as follows:
divisors: 4 5
reminders: 1 4
leave the right-most top number 5
multiply 4 and 4, then add 1 which is down below the divisor 4; you will get 17
Now, add this 17 to the product of 4, 5, (divisors) and k is a constant
20k+17 , this is the number we need
when k=1, the number is 37
when 37 is divided by 5, the remainder is 2.
when 7 is divided by 4, the remainder is 3.

Approach Solution 3:
The first such number of the possible sequence be x. then x= 4y+1, y be the dividend. then, now, y=  5z+4

=> x= 4(5z+4)+1=20z+17. to determine the first number of the series, set z= 1.

Then x=37. successively divided by 5 and 4 we get remainders 2 and 3. 

“A number when divided successively by 4 and 5 leaves remainder 1 and 4 respectively.”- is a topic of the GMAT Quantitative reasoning section of GMAT. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. GMAT Quant practice papers improve the mathematical knowledge of the candidates as it represents multiple sorts of quantitative problems.
 

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