Question

What is the tens digit of positive integer $x?$
$(1)$ $x$ divided by $100$ has a remainder of $30.$
$(2)$ $x$ divided by $110$ has a remainder of $30.$
A. Statement $(1)$ alone is sufficient, but statement $(2)$ alone is not sufficient.
B. statement $(2)$ alone is sufficient, but statement $(1)$ alone is not sufficient.
C. Both statements together are sufficient, but neither statement alone is sufficient.
D. Each statement alone is sufficient.
E. Statements $(1)$ and $(2)$ together are not sufficient.

Answer 1

Hint: Here we will take both the given statements one by one and will check for the tens digit. Tens digit is the second place next to the unit place moving from right hand side to the left hand side of the number. Apply the properties of the division over here.

Complete step by step answer:
Take the first statement, given that when we divide any number by $100$ has a remainder of $30.$ So, for example $430$ and when it is divided by $100$,
$ \Rightarrow \dfrac{{430}}{{100}}$ gives $30$ has the remainder.
So, tens digit in any number should be 3. Hence, statement one is sufficient.

Now, similarly take the second given statement.Given that when we divide any number by $110$ has a remainder of $30.$ So, for example $470$ and when it is divided by $110$,
$ \Rightarrow \dfrac{{470}}{{110}}$ gives $30$ has the remainder.
Let us take another example –
$470$and when it is divided by $110$,
$ \Rightarrow \dfrac{{580}}{{110}}$ gives $30$ has the remainder.
So, tens digit in the above two examples is not and has and has the tens place and so the tens place is not uniquely determined and so the statement second is insufficient.

Hence, option (A) is the correct answer.

Note: Always remember the division algorithm properly which states that dividend is equal to the sum of the remainder with the product of the divisor and the quotient. Be good in multiples and the division and apply it accordingly. Always cross-check the values by using the division algorithm.