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In a question on division, the divisor is 7 times the quotient and 3 times the remainder. If the remainder is 28, then the dividend is: \[\]

Last updated date: 13th Jun 2024
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Hint: We make equations $d=7\times q=7q,d=3\times r=3r$ with the information given in the question where $d$ is the divisor, $q$ is the quotient and $r$ is the remainder. We put $r=28$ in the second equation to find $d$ and put the obtained value of $d$ in the first equation to get $q$. We put $d,q,r$ in the Euclid’s lemma of division to find dividend $n=dq+r$ .\[\]

Complete step by step answer:
We know that in arithmetic operation of division the number we are going to divide is called dividend, the number by which divides the dividend is called divisor. We get a quotient which is the number of times the divisor is of dividend and also remainder obtained at the end of division. \[\]If the number is $n$, the divisor is $d$, the quotient is $q$ and the remainder is $r$, they are related by the following equation,
Here the divisor can never be zero. The above relation is called Euclid’s division Lemma.
We are given in the question that the divisor is 7 times the quotient and 3 times the remainder. We know that in mathematics ‘times’ means multiplication. We have,
  & d=7\times q=7q........\left( 1 \right) \\
 & d=3\times r=3r.........\left( 2 \right) \\
We are further given in the question the remainder is 28. Let us put $r=28$ in equation (2) and have the dividend as
\[d=3r=3\times 28=84\]
We put $d=84$ in equation and have,
We divide both side of the above equation by 7 to have,
  & \dfrac{84}{7}=\dfrac{7q}{7} \\
 & \Rightarrow 12=q \\
We put the values of obtained divisor$d=84$, quotient $q=12$ and remainder $r=28$ in Euclid’ division lemma to have the dividend as
\[n=dq+r=84\times 12+28=1008+28=1036\]

So, the correct answer is “Option C”.

Note: We note that ‘lemma’ means a small result accompanying a theorem. We can alternatively assume the divisor as an unknown variable $x$ and solve for $x$. We also note that in Euclid’s lemma $n=dq+r$ we always have $0\le r<\left| b \right|$ where $\left| b \right|$ is the absolute value of $b$.