Question

A number consists of two digits. When it is divided by the sum of its digits, the quotient is 6 with no remainder, when the number is diminished by 9, the digits are reversed. Find the number?

Answer 1

Hint: To solve this type of questions you must be aware of \[Dividend = Divisor \times Quotient + Remainder\], two digit number is denoted by $10x + y$ and sum of the number is denoted by $x + y$ so according to the question, substitute the values in the form and find the values of $x$ and $y$.

Complete step-by-step answer:
So, we know that \[Dividend = Divisor \times Quotient + Remainder\],and by this
$
   \Rightarrow Dividend = 10x + y \\
   \Rightarrow Divisor = x + y \\
   \Rightarrow Quotient = 6 \\
   \Rightarrow \operatorname{Re}mainder = 0 \\
 $
So, now on solving we will get a linear equation,
$
   \Rightarrow 10x + y = (x + y) \times 6 + 0 \\
   \Rightarrow 10x + y = 6x + 6y \\
   \Rightarrow 4x = 5y \\
   \Rightarrow 4x - 5y = 0.......\left( 1 \right) \\
 $
Additional information says that when the number is subtracted by 9 the resultant is the reverse of the number.
So,
$
   \Rightarrow 10x + y - 9 = 10y + x \\
   \Rightarrow 9x - 9 = 9y \\
   \Rightarrow 9x - 9y = 9 \\
   \Rightarrow x - y = 1........\left( 2 \right) \\
    \\
 $
On Solve equation (1) and equation (2)
From equation (2), we have $x = y + 1$
Substitute in equation (1), we get
 $
   \Rightarrow 4(y + 1) = 5y \\
   \Rightarrow 4y + 4 = 5y \\
   \Rightarrow y = 4 \\
 $
Substitute $y = 4$ in equation (2) we get
 $
   \Rightarrow x - y = 1 \\
   \Rightarrow x - 4 = 1 \\
   \Rightarrow x = 5 \\
 $
So, the two digit number is $10x + y = 10(5) + 4 = 54$.

Note: Two digit number is denoted by the form $10x + y$, and also take care of the condition \[Dividend = Divisor \times Quotient + Remainder\], take care what comes in dividend, divisor, quotient and remainder.