Question

64329 is divided by a certain number. While dividing, the number 175, 114 and 213 appear as three successive remainders. The divisor is?
A) 184
B) 224
C) 234
D) 296

Answer 1

Hint:
Here, we will show this situation as a long division and find the missing number by subtracting the successive remainders from the previous one as shown. Finding the H.C.F. of those numbers will give us the required divisor.

Complete step by step solution:
In order to find the divisor, first of all, we will write the given situation in terms of long division:
Where, it is given that the Dividend, i.e. the number which is divided is 64329
Also, while dividing the number, the numbers 175, 114 and 213 appear as three successive remainders.
Hence, we will show this information as:
\[64329 \div x = 1752\]……………………\[\left( 1 \right)\]
\[1752 \div y = 1149\]………………………….\[\left( 2 \right)\]
\[1149 \div z = 213\]………………………………..\[\left( 3 \right)\]
Hence, from the above division we can find the number at \[\left( 1 \right)\] as:
\[643 - 175 = 468\]
Similarly, the number at \[\left( 2 \right)\] is the difference of:
\[1752 - 114 = 1638\]
And, the number at \[\left( 3 \right)\] is:
\[1149 - 213 = 936\]
Now, the numbers found by us are: 468, 1638, 936
These numbers can be factorized as:
\[468 = {2^2} \times {3^2} \times 13\]
\[936 = {2^3} \times {3^2} \times 13\]
And, \[1638 = 2 \times {3^2} \times 7 \times 13\]
Now, the H.C.F. of these three numbers will give us the required divisor.
Hence, the numbers which are common in all the three numbers with the lowest possible power, i.e. the H.C.F. of these three numbers \[ = 2 \times 3 \times 3 \times 13 = 18 \times 13 = 234\]
Therefore, the required divisor is 234.

Hence, option C is the correct answer.

Note:
Here, successive remainder means the remainder obtained in three consecutive divisions. We should not get confused between the divisor and dividend. A divisor is a number by which the other number will be divided, whereas a dividend is the number which is to be divided. Here, we are given a dividend and not a divisor. HCF or the Highest Common Factor of two or more numbers is the greatest factor that divides all the numbers.