Question

When 31513 and 34369 are divided by a certain three digit number, the remainders are equal then the remainder is
A) 86
B) 97
C) 374
D) 113

Answer 1

Hint:
Here we will first assume the divisor and the remainder to be some variable. Then we will write the numbers in expanded form of the divisor, quotient and remainder. Then we will subtract the numbers and find the possible values of the 3 digit divisors. Then we will find the remainder by simple division.

Complete Step by Step Solution:
Given numbers are 31513 and 34369.
Let the divisor of the numbers be \[d\] and the remainder be \[r\].
Now we will write both the numbers in the expanded form of the divisor, quotient and the remainder. Therefore, we get
\[31513 = dn + r\]…………………………….\[\left( 1 \right)\]
\[34369 = dm + r\]…………………………..\[\left( 2 \right)\]
Here \[n\] and \[m\] are the quotients of the numbers.
Now we will subtract the equation \[\left( 1 \right)\] from the equation \[\left( 2 \right)\]. Therefore, we get
\[34369 - 31513 = dm + r - \left( {dn + r} \right)\]
Subtracting the like terms, we get
\[ \Rightarrow 2856 = dm + r - dn - r\]
Now by simplifying it, we get
\[ \Rightarrow d\left( {m - n} \right) = 2856\]
Now we will make all the possible 3 digit multiples for the above equation, we get
\[\begin{array}{l}d\left( {m - n} \right) = 2856 = 119 \times 24\\d\left( {m - n} \right) = 2856 = 238 \times 12\\d\left( {m - n} \right) = 2856 = 357 \times 8\\d\left( {m - n} \right) = 2856 = 476 \times 6\\d\left( {m - n} \right) = 2856 = 714 \times 4\end{array}\]
So, the possible values of the divisor are \[d = 119,238,357,476,714\] and by dividing the given numbers by these possible values of the divisors we get a remainder of only 97.
For example: \[\dfrac{{31513}}{{238}} = 132\dfrac{{97}}{{238}}\] and \[\dfrac{{34369}}{{238}} = 144\dfrac{{97}}{{238}}\]

Hence when 31513 and 34369 are divided by a certain three digit number, the remainders are equal then the remainder is 97.

Note:
We should know that the division is the operation in which the dividend is divided by the divisor to get the quotient along with some remainder. So, the general formula of the division operation is
\[{\rm{Dividend}} = \left( {{\rm{Divisor}} \times {\rm{Quotient}}} \right) + {\rm{Remainder}}\]
Dividend is the term or number which is to be divided. Divisor is the term or number which we divide by. Quotient is the term or number that is the answer of the division operation and remainder is the term which is left when a division operation is performed.