The number of ways of selecting two numbers $a$ and $b, a \in\{2,4,6, \ldots , 100\}$ and $b \in\{1,3,5, \ldots , 99\}$ such that 2 is the remainder when $a+b$ is divided by 23 is
- 54
- 108
- 268
- 186
The Correct Option is B
Solution and Explanation
\(a∈{2,4,6,8,10,….,100} \)
\(b∈{1,3,5,7,9,……,99} \)
Now, \(a+b∈{25,71,117,163}\)
(i) \(a+b=25\), no. of ordered pairs \((a, b)\) is \(12\)
(ii) \(a+b=71\), no. of ordered pairs \((a, b)\) is \(35\)
(iii) \(a+b=117\), no. of ordered pairs \((a, b)\) is \(42\)
(iv) \(a+b=163\), no. of ordered pairs \((a, b)\) is \(19\)
\(∴ total =108 \;pairs\)
\(\text{Therefore, the correct option is (B):} 108\)
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Concepts Used:
Sets
Set is the collection of well defined objects. Sets are represented by capital letters, eg. A={}. Sets are composed of elements which could be numbers, letters, shapes, etc.
Example of set: Set of vowels A={a,e,i,o,u}
Representation of Sets
There are three basic notation or representation of sets are as follows:
Statement Form: The statement representation describes a statement to show what are the elements of a set.
- For example, Set A is the list of the first five odd numbers.
Roster Form: The form in which elements are listed in set. Elements in the set is seperatrd by comma and enclosed within the curly braces.
- For example represent the set of vowels in roster form.
A={a,e,i,o,u}
Set Builder Form:
- The set builder representation has a certain rule or a statement that specifically describes the common feature of all the elements of a set.
- The set builder form uses a vertical bar in its representation, with a text describing the character of the elements of the set.
- For example, A = { k | k is an even number, k ≤ 20}. The statement says, all the elements of set A are even numbers that are less than or equal to 20.
- Sometimes a ":" is used in the place of the "|".