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 (factors of 585) and got the remainder 4, 8 and 12. If he had divided the number by 585, then find out the remainder.

Answer 1

Hint: To solve the above question, we will first assume that the unknown number which is divided by 5, 9 and 13 is n. Then we will use the property of division which is Dividend = Divisor \[\times \] Quotient + Remainder. By using this property, we will assume that when n is divided by 5, the quotient will be x. When x will be divided by 9, the quotient will be y. When y will be divided by 13, the quotient will be z. This, we will get 3 equations. From these three equations, we will obtain a relation between n and z. When we will derive a relation between n and z, we get, n = 585z + r. From there, we will find the value of r.
Complete step by step solution:
To start with, we will assume that the number which is divided by 5, 9 and 13 is n. Now, we will use the property of the division that is given by,
Dividend = Divisor \[\times \] Quotient + Remainder
Now, we have to find the remainder when n is divided by 585. Let the quotient be ‘q’ and the remainder be ‘r’ in this case. Thus, we will get,
\[n=585\times q+r\]
\[\Rightarrow n=585q+r.....\left( i \right)\]
Now, it is given in the question that n is divided by 5 and the quotient obtained is divided by 9 and again the quotient obtained is divided by 13. We will now assume that when n is divided by 5, the quotient is x. So, we will get,
\[n=5x+{{r}_{1}}\]
where \[{{r}_{1}}\] is the remainder and it is given that \[{{r}_{1}}=4.\] So, we have,
\[n=5x+4.....\left( ii \right)\]
Now, when x is divided by 9, we will assume that the quotient obtained is y and the remainder is 8. So, we will get,
\[x=9y+8....\left( iii \right)\]
Now, y is divided by 13 and the remainder obtained is 12. We will assume that the quotient is z. So, we will get,
\[y=13z+12....\left( iv \right)\]
Now we will put the value of y from (iv) to (iii). Thus, we will get,
\[\Rightarrow x=9\left( 13z+12 \right)+8\]
\[\Rightarrow x=117z+108+8\]
\[\Rightarrow x=117z+116....\left( v \right)\]
Now, we will put the value of x from (v) to (ii). Thus, we will get,
\[\Rightarrow n=5\left( 117z+116 \right)+4\]
\[\Rightarrow n=585z+580+4\]
\[\Rightarrow n=585z+584.....\left( vi \right)\]
Now, we will compare the equations (vi) and (i). Thus, we will get, q = z and r = 584.
Thus, the remainder is 584. Hence when n is divided by 585, the remainder obtained is 584.

Note: The alternate method to solve the above question is shown below:
We can notice that when n is divided by the numbers, the remainder is 1 less than the quotient, i.e.
N = 5x + 4
N = 9y + 8
N = 13z + 12
We can write the above equations as
N + 1 = 5 (x + 1)
N + 1 = 9 (y + 1)
N + 1 = 13 (z + 1)
Thus, N + 1 is divisible by 5, 9 and 13. So, N + 1 will be divisible by 585. So, we can say that
\[N+1=585z\]
\[\Rightarrow N=585z-1\]
\[\Rightarrow N=585\left( z-1 \right)+585-1\]
\[\Rightarrow N=585{{z}^{'}}+584\]
Hence, the remainder is 584.