In dividing a number by 585, a student employed the method of short di

The two solutions posted above are based on the assumption that there is a consistent relationship between each of the divisors and the corresponding remainder (divisor is 1 more than remainder in each case). But what if this is not the case? To illustrate this, I have composed an identical problem but with different values for divisors and remainders:

"A certain number is successively divided by 2, 3 & 4 (the factors of 24) and the remainders are 1, 2 & 0 respectively. What will the remainder be if the number is divided outright by 24?"

As you can see, there is no discernible pattern between the divisors and remainders. My solution (which can be applied to all problems of this nature) is:

Let N be the number and Q1, Q2 & Q3 be the quotients for the first, second and third divisions respectively.
2Q1 + 1=N ........ (i)
3Q2 + 2=Q1 ...... (ii)
4Q3 + 0=Q2 .......(iii)
We also know that 24Q + R=N where Q is the quotient and R is the remainder when the number N is divided outright by 24. Now, it is a fact that when a number 'n' is successively divided by the factors of another number 'd', the last quotient is the same as the quotient when 'n' is divided outright by 'd'. This can easily be proved algebraically or verified by some simple examples (for instance, n=18, d=6, d1=2 and d2=3).
Therefore, 24Q3 + R=N
From (ii) and (iii), we can deduce that Q3=(Q1-2)/12
Thus, 24(Q1-2)/12 +R=N ......(iv)
So, from (iv) and (i), we have: 2(Q1-2) + R=2Q1 + 1; R=5. (N=53, Q1=26, Q2=8 and Q3=2)

This approach will also yield the correct result (remainder=584) when applied to the original problem. Hope I have been able to provide a clear explanation.

While the above methods are crisp, clear and to-the-point, I would like to share the method of solving backwards from answer choices, which may be helpful in case you are not able to think of any of the above approaches.

Let the unknown number be \(x\)

'If he had divided the number by 585' - Then the equation would be \(x=585q+r,\) where \(q\) is the quotient and \(r\) is the remainder. And hence \(r\) would be one of the 5 answer choices. For sake of simplicity lets assume \(q\) to be equal to 1.

While checking the answer choices we need to check if the resulting number leaves a remainder of 4, 8 and 12 when divided by 5, 9 and 13

A. \(585*1 + 24 = 609\) - Does not leave 8 as the remainder when divided by 9 as 603 is the closest multiple to 609. Eliminate
B. \(585*1 + 144 = 729\) - Is divisible by 9 and hence leaves 0 as the remainder instead of 8. Eliminate
C. \(585*1+ 288 = 873\) - The closest multiple of 5 to 873 is 870 which leaves remainder of 3 instead of 4. Eliminate
D. \(585*1 + 292 = 877\) - Again, the closest multiple of 5 to 875 is 877 which leaves remainder of 2 instead of 4. Eliminate

This leaves us with only 1 choice and we can mark it without making any calculations.

Ans. E

We see that each remainder is 1 less than its respective divisor (4 is 1 less than 5, 8 is 1 less than 9, and 12 is 1 less than 13). Therefore, when the number is divided by 585, the remainder will be 1 less than 585, which is 584.

Answer: E

Can you explain why it would be so?

Response:

Let X be the number. X is divided successively by 5, 9 and 13 and the remainders are 4, 8 and 12, respectively. This can be expressed algebraically as follows:

X = 5k + 4

k = 9s + 8

s = 13t + 12

Here, k, s, and t are quotients obtained from the divisions. Let’s add 1 to each side of the equality X = 5k + 4:

X + 1 = 5k + 4 + 1 = 5k + 5 = 5(k + 1)

Let’s substitute 9s + 8 for k:

X + 1 = 5(k + 1) = 5(9s + 8 + 1) = 5(9s + 9) = 5 * 9(s + 1)

Let’s substitute 13t + 12 for s:

X + 1 = 5 * 9(13t + 12 + 1) = 5 * 9(13t + 13) = 5 * 9 * 13(t + 1)

We see that X + 1 is divisible by 5 * 9 * 13 = 585. Since X + 1 is divisible by 585, the remainder when X is divided by 585 must be 584.

This shows that when the remainders are 1 less than the divisors, adding 1 to the number will result in a multiple of all the divisors. Thus, when the number is divided by the product of the divisors, the remainder will be 1 less than the number.

Given that a student divided a number by 585 using the method of short division. And he successively divided the number by 5, 9 and 13 (factors 585) and got the remainders 4, 8, 12 respectively. And we need to find the remainder when the he divided the number by 585

He divided 585 by 5, 9 and 13 successively to get the remainders of 4, 8 and 12. Let's try to find the number by going in the reverse direction

Divided a number by 13 to get get 12 remainder

Theory: Dividend = Divisor*Quotient + Remainder

Number -> Dividend
13 -> Divisor
a -> Quotient (Assume)
12 -> Remainders
=> Number which was divided by 13 = 13*a + 12 = 13a + 12

Divided something by 9 to get 13a + 12 as quotient and 8 as remainder
=> Number which was divided by 9 = 9*(13a + 12) + 8 = 117a + 108 + 8 = 117a + 116

Divided something by 5 to get 117a + 116 as quotient and 4 as remainder
=> Actual Number = 5*(117a + 116) + 4 = 585a + 580 + 4 = 585a + 584

Remainder when Actual Number is divided by 585

585a + 584 when divided by 585 will give 584 as remainder

So, Answer will be E
Hope it helps!

Watch the following video to learn the Basics of Remainders

Given: In dividing a number by 585, a student employed the method of short division. He divided the number successively by 5, 9 and 13 (factors 585) and got the remainders 4, 8, 12 respectively.

Asked: If he had divided the number by 585, the remainder would have been

Let the number be x.

x = 5k + 4
k = 9m + 8
m = 13n + 12

x = 5(9(13n+12)+8)+4 = 585n + 540 + 40 + 4 = 585n + 584

IMO E

Let the number be n.
(((5n+4)9+8)13+12)=585n+584. Therefore, the remainder is 584(E).