Question

When a two digit number is divided by the sum of the digits of the number,the quotient becomes 9 . But when the digits are reversed ,the quotient by dividing this new number by the sum becomes 2 . Find the number

Answer 1

Answer:

81

Step-by-step explanation:

Let the ten's digit be and unit's digit be y.

Original number = 10+y

Sum of the digits = +y

A.T.P.

=) (10+y)/(+y) = 9

=) 10+y = 9+9y

=) =8y

Reversed number = 10y+

A.T.P.

=) (10y+)/(+y)=2

=) 10y+=2+2y

=) 8y=

Now,

A digit must be between 0 to 9

If y is 0 then will also be zero. Hence, number = 0 (not possible)

If y is 1 then will be 8. Hence, number = 81 (possible)

If y is 2 then will be 16 (not possible)

Rest continues the same.....

Answer 2

Question:-

When a two digit number is divided by the sum of the digits of the number, the quotient becomes 9. But when the digits are reversed, the quotient obtained on dividing this new number by the sum becomes 2. Find the number.

Answer:-

$\displaystyle\Large\boxed {\sf {81}}$

Solution:-

The two digit number = $\displaystyle\sf {10x+y}$

Given,

• The two digit number gives quotient 9 on division by the sum of the digits. → (i)

• The two digit number formed by reversing the digits gives quotient 2 on division by the sum of the digits. → (ii)

Assume no remainder is obtained.

By (i),

$\displaystyle\longrightarrow\sf{\dfrac {10x+y}{x+y}=9}$

$\displaystyle\longrightarrow\sf{10x+y=9x+9y}$

$\displaystyle\longrightarrow\sf{x=8y}$

The original two digit number will be,

$\displaystyle\longrightarrow\sf{10x+y=10\times8y+y}$

$\displaystyle\longrightarrow\sf{10x+y=80y+y}$

$\displaystyle\longrightarrow\sf{10x+y=81y}$

The possible value for 'y' is 1 since $\displaystyle\sf {10x+y=81y}$ is a two digit number.

If $\displaystyle\sf {y\geq2}$ then $\displaystyle\sf {10x+y}$ will be a three digit number or more.

Since $\displaystyle\sf {81y}$ is a two digit number,

$\displaystyle\longrightarrow\sf{81y\ \textless\ 100}$

$\displaystyle\longrightarrow\sf{y\ \textless\ \dfrac {100}{81}}$

$\displaystyle\longrightarrow\sf{y\ \textless\ 1.23456789}$

$\displaystyle\Longrightarrow\sf{y=1}$

Hence,

$\displaystyle\Large\boxed {\sf {10x+y=81}}$