$1 x 35$ is divisible by $9$ if $x$ $=$ _______.

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Solution

It is given that,
$1 x 35$ is divisible by $9$ if $x$ $=$ _______.
Here,
$1 x 35$ is Four digit number and $x$ is a single-digit number.
Note :
A number is divisible by $9$ if the sum of its digits is also divisible by $9$ .
and
A number is divisible by $9$ means it is a multiple of $9$ .
So Consider,
The sum of the digits is
$1 + x + 3 + 5$ $=$ A number multiple of $9$ .
$x + 9$ $=$ A number multiple of $9$ .
According to the question, $x$ is a single-digit number.
Here, There are two possible cases for $x$ i.e $x$ $=$ $0$ or $x$ $=$ $9$ .
Case i :
If $x$ $=$ $0$ then
$x + 9$ $=$ $0$ $+$ $9$
$=$ $9$ . ( So, $9$ is divisible by $9$ ).
Case ii :
If $x$ $=$ $9$ then
$x + 9$ $=$ $9$ $+$ $9$
$=$ $18$ . ( So, $18$ is divisible by $9$ ).
Therefore, it is observed that there are two possible single digit values for $x$ to get a four digit number divisible by $9$ from above two cases.
Hence, $x$ $=$ $0$ or $9$ .

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