EXERCISE - 2.3

In base $10,$ the number $2013$ ends in the digit $3$. In base $9,$ on the other hand, the same number is written as $\left(2676\right) 9$ and ends in the digit $6.$ For how many positive integers $b$ does the base-$b$ representation of $2013$ end in the digit $3$?

Real numbers $x$ and $y$ satisfy the equation $x^2+y^2=10x-6y-34.$ What is $x+y$ ?

The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. How many integers less than $2013$ but greater than $1000$ share this property?

The number $2013$ is expressed in the form $2013=\frac{\left(a_1!\right)\left(a_2!\right)\dots \left(a_m!\right)}{\left(b_1!\right)\left(b_2!\right)\dots \left(b_n!\right)}$

where $a_1\ge a_2\ge \dots \ge a_m$ and $b_1\ge b_2\ge \dots b_n$ are positive integers and $a_1+b_1$ is as small as possible. What is
$\left|a_1-b_1\right|?$

Two integers have a sum of $26.$ When two more integers are added to the first two integers the sum is $41.$ Finally, when two more integers are added to the sum of the previous four integers the sum is $57.$ What is the minimum number of even integers among the $6$ integers?

How many ordered pairs of positive integers $\left(M,N\right)$ satisfy the equation $\frac{M}{6}=\frac{6}{N}$?

Geeta and Babeeta play the following game. An integer between $0$ and $999$, inclusive, is selected and given to Geeta. Whenever Geeta receives a number, she doubles it and passes the result to Babeeta. Whenever Babeeta receives a number, she adds $50$ to it and passes the result to Geeta. The winner is the last person who produces a number less than $1000.$ Let $N$ be the smallest initial number that results in a win for Geeta. What is the sum of the digits of $N$?

A majority of the $30$ students in Mrs. Bhatnagar's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1.$ The cost of a pencil in paisa was greater than the number of pencils each student bought, and the total cost of all the pencils was $₹ 17.71.$ What was the cost of a pencil in paisa?