Question

On dividing a positive integer n by 9. We get 7 as remainder. What will be the remainder if \[\left( {3n - 1} \right)\] is divided by 9?
(A) 1
(B) 2
(C) 3
(D) 4

Answer 1

Hint: According to the given question, apply Euclid’s algorithm that is on dividing n by 9. Then, substitute the values in the formula that is Dividend = divisor \[ * \] quotient \[ + \]remainder.
Hence, calculate the remainder by making the left hand side equal to \[\left( {3n - 1} \right)\]

Formula used:
Here, we use the formula of Euclid’s algorithm that is Dividend = divisor \[ * \] quotient \[ + \] remainder.

Complete step-by-step answer:
Let us take q to be the quotient.
It is given that the remainder = 7
On applying Euclid’s algorithm that is on dividing n by 9 we have Dividend = divisor \[ * \] quotient \[ + \]remainder.
So, substituting the values of dividend which is equals to n, divisor which is equals to 9, quotient equals to q and remainder is equals to 7.
\[n{\rm{ }} = {\rm{ }}9q{\rm{ }} + {\rm{ }}7\;\]
Multiply 3 with all the equation we get,
\[ \Rightarrow 3n{\rm{ }} = {\rm{ }}27q{\rm{ }} + {\rm{ }}21\;\]
Subtracting -1 from both left and right hand side. So,
\[ \Rightarrow 3n-1{\rm{ }} = {\rm{ }}27q{\rm{ }} + {\rm{ }}20\;\]
Make in the form of 9 \[ * \] quotient \[ + \] remainder.
\[ \Rightarrow 3n-1{\rm{ }} = {\rm{ }}9 \times 3q{\rm{ }} + {\rm{ }}9 \times 2{\rm{ }} + {\rm{ }}2\]
Taking out all the common values from right hand side we get
\[ \Rightarrow 3n-1{\rm{ }} = {\rm{ }}9 \times \left( {3q{\rm{ }} + {\rm{ }}2} \right){\rm{ }} + {\rm{ }}2\;\]
Therefore, when (3n-1) is divided by 9, we get the remainder 2.
Hence, option (B) 2 is correct.

Additional Information:
Euclid’s algorithm is used for calculating positive integers values of the required question. It is basically the HCF of two positive integers that can be a and b.


Note: To solve these types of questions, we should apply Euclid’s algorithm in the given statement.
Then, calculate the values of dividend, divisor, and quotient and remainder carefully. Hence, calculate the given question by putting the required values and simplify according to the ask of the question.