Question

A number was divided successively in order by 4, 5, and 6. The remainders were 2, 3, and 4. Then, find out the number.

Answer 1

Hint: Assume that the original number is x. In the first case, x is divided by 4 and the remainder is equal to 2. Let us assume that the quotient when the number x is divided by 4 be p. Here, the divisor is equal to 4, the quotient is equal to p, the remainder is equal to 2, and the dividend is x. Now, use the formula, \[\text{Dividend=Divisor }\!\!\times\!\!\text{ Quotient+Remainder}\] . In the second case, p is divided by 5 and the remainder is equal to 3. Let us assume that the quotient when the number p is divided by 5 be q. Here, the divisor is equal to 5, the quotient is equal to q, the remainder is equal to 2, and the dividend is p. Now, use the formula, \[\text{Dividend=Divisor }\!\!\times\!\!\text{ Quotient+Remainder}\] . In the third case, q is divided by 6 and the remainder is equal to 3. Let us assume that the quotient when the number q is divided by 6 be r. Here, the divisor is equal to 6, the quotient is equal to r, the remainder is equal to 3, and the dividend is q. Now, use the formula, \[\text{Dividend=Divisor }\!\!\times\!\!\text{ Quotient+Remainder}\] . Now, express x in terms of r. Then, put r = 1 and solve it to get the value of x.

Complete step-by-step answer:
According to the question, it is given that a number was divided successively in order by 4, 5, and 6. The remainders were 2, 3, and 4.
Here, we have three cases.
Let us assume that the original number is x.
In the \[{{1}^{st}}\] case, it is given that the original number is divided by 4 and the remainder is equal to 2.
Let us assume that the quotient when the number x is divided by 4 is p.
So, the divisor is equal to 4, the quotient is equal to p, the remainder is equal to 2, and the dividend is x.
Dividend = x ……………………………(1)
Divisor = 4 ……………………………..(2)
Remainder = 2 ………………………………..(3)
Quotient = p ……………………………………(4)
We know the formula, \[\text{Dividend=Divisor }\!\!\times\!\!\text{ Quotient+Remainder}\] ………………………………….(5)
Now, from equation (1), equation (2), equation (3), equation (4), and equation (5), we get
\[\Rightarrow x=4\times p+2\]
\[\Rightarrow x=4p+2\] ………………………………(6)
In the \[{{2}^{nd}}\] case, it is given that the quotient obtained by dividing the original number is divided by 5 and the remainder is equal to 3.
From equation (4), we have the quotient when the original number is divided by 5.
Let us assume that the quotient when the number p is divided by 5 is q.
So, the divisor is equal to 5, the quotient is equal to q, the remainder is equal to 3, and the dividend is p.
Dividend = p ……………………………(7)
Divisor = 5 ……………………………..(8)
Remainder = 3 ………………………………..(9)
Quotient = q ……………………………………(10)
Now, from equation (5), equation (7), equation (8), equation (9), and equation (10), we get
\[\Rightarrow p=5\times q+3\]
\[\Rightarrow p=5q+3\] ………………………………(11)
In the \[{{2}^{nd}}\] case, it is given that the number q is divided by 6 and the remainder is equal to 4.
Let us assume that the quotient when the number q is divided by 5 is r.
So, the divisor is equal to 6, the quotient is equal to r, the remainder is equal to 4, and the dividend is q.
Dividend = q ……………………………(12)
Divisor = 6 ……………………………..(13)
Remainder = 4 ………………………………..(14)
Quotient = r ……………………………………(15)
Now, from equation (5), equation (12), equation (13), equation (14), and equation (15), we get
\[\Rightarrow q=6\times r+4\]
\[\Rightarrow q=6r+4\] ………………………………(16)
Now, from equation (11) and equation (16), we get
\[\Rightarrow p=5\left( 6r+4 \right)+3\]
\[\Rightarrow p=30r+23\] …………………………..(17)
Now, from equation (6) and equation (17), we get
\[\Rightarrow x=4\left( 30r+23 \right)+2\]
\[\Rightarrow x=120r+94\] …………………………..(18)
The original number x is of the form \[\left( 120r+94 \right)\] ………………………………(19)
Putting r = 1 in equation (19), we get
\[\begin{align}
  & \Rightarrow x=120\times 1+94 \\
 & \Rightarrow x=120+94 \\
 & \Rightarrow x=214 \\
\end{align}\]
Hence, the original number is 214.

Note:In this question, one might think that the original number is divided by 4, then by 5, and then by 6. This is wrong because the original number is divided by 4 and then the quotient obtained after dividing by 4 is divided by 5. Then, the quotient which is obtained after dividing by 5 is divided by 6.