Question: In dividing a number by 585, a student employed the method of short division. He divided the number successively by 5, 9, and 13 (factor 585) and got the remainders 4, 8, 12 respectively. If he had divided the number by 585, the remainder would have been:

A. 24
B. 144
C. 288
D. 292
E. 584

Answer: E

Approach Solution (1):
If z was the last number to be successively divided by 13, it yielded the remainder of 12. Thus, a remainder of 12 indicates that the value of 1 less than being divisible by 13.
Hence, it is calculated as:
Z = 13 x 1 + 12 = 25
The last number is 25.
Let the previous number is denoted as y
Value of y = 9 x z + 8
= 9 x 25 + 8
= 233
The value of the earlier number is 233
Let the original number be x
The value of x is calculated as = 5 × y + 4
= 5 × 233 + 4
= 1169
Thus, the original number is 1169
When 1169 was divided by 585, it would yield a remainder of 584
Correct option: E

Approach Solution (2):
From L.C.M and H.C.F methods
We have two formulas
i.e
1. To find the greatest number that will divide by x,y&z leaving remainders a, b, c, respectively.
Required no. = H.C.F of(x-a), (y-b) & (z-c).
2. To find the least number which when divided by x, y, z leaves the remainder a, b, c, respectively.
It is always observed that (x-a) = (y-b) = (z-c) = k
Required no. = (L.C.M of x, y & z) - k
But in question, we have not asked for find out the least or greatest number, So by ignoring that least and greatest number, we have to notice on the given dividends and remainders.
Here,
we have 3 dividends and 3 remainder i.e,
Dividends =5, 9, 13.
Remainders =4, 8, 12.
Now, we have to subtract the remainders of the respective dividends to check if the differences are same then here we apply the L.CM rule but if the differences are different then we will apply the H.C F rule.
But here the differences between every dividend and remainder is same i.e,
(5 - 4) = (9 - 8) = (13 - 12) = (k) 1
As the differences are same here we apply the formula 1 to find the remainder.
Required number = (L.C.M of x, y & z) - k
= (L.C.M of 5, 9 & 13) - ,
= 585 - 1
= 584 (remainder) answer
Correct option: E

Approach Solution (3):
Let p be the number
It is given that
p/5 = q, remainder = 4
p/9 = r, remainder = 8
r/13 = 1, remainder = 12
Then
Formula: Dividend = Divisor * Quotient + Remainder
r = (1 * 13) + 12 = 13 + 12 = 25
q = 9r + 8 = (9 * 25) + 8 = 225 + 8 = 233
p = 5q + 4 = (5 * 233) + 4 = 1165 + 4 = 1169
Now, we have to find the remainder when the number is divided by 585 that is:
p/585 = 1169/585 = 1
Hence, the remainder = 584
Correct option: E

“In dividing a number by 585, a student employed the method of short division. He divided the number successively by 5, 9, and 13 (factor 585) and got the remainders 4, 8, 12 respectively. If he had divided the number by 585, the remainder would have been:”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “GMAT Official Guide Quantitative Review”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

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