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While preparing the Number System for CAT exam, the very first thing that you should master is the types of numbers used in CAT Number System Questions. It is one concept which is not only used in number system but is also used in almost all the topics of quantitative aptitude. Also, you can never ignore the type of numbers given in a particular question because that can change the whole answer. We will discuss this by taking the examples.
Let us first of all discuss the different type of numbers.
1. Natural Numbers: The numbers 1, 2, 3, 4, …….. are called natural numbers. These are also called counting numbers.
2. Whole numbers: The set of natural numbers including 0 is called the set of natural numbers. Therefore, 0, 1, 2, 3, 4,…….. are all whole numbers.
Therefore, every natural number is a whole number but the converse is not true, i.e., every whole number need not to be a natural number.
3. Integers: If we take the whole numbers and their negatives as well, then the set that we obtain is called the sets of integers. Therefore, ………. – 4, -3, – 2, -1, 0, 1, 2, 3, 4, ….. are all integers. Remember that the integers do not contain any fractions.
Therefore, every whole number is an integer but the converse is not true, i.e., every integer need not to be a whole number.
4. Rational numbers: The numbers which can be written as p/q, where ‘p’ and ‘q’ are integers and ‘q’ ≠ 0 are called rational numbers. E.g., 1/2, – 4/5, 11/8 etc. are rational numbers.
Also note that 5 can be written as 5/1 and – 8 can be written as – 8/1, which are also of the form p/q. Therefore, every integer is also a rational number but the converse is not true i.e., every rational number need not to be an integer.
There are two other types of numbers given below, which are not directly of p/q form but can be converted into this form. So, they are also considered as rational numbers.
(a) Terminating decimals: Terminating decimals are also rational numbers as they can be converted into p/q form. E.g., 0.25 is a terminating decimal. It can be written as 25/100 = 1/4.
(b) Non terminating recurring decimals: Non terminating decimals are the ones which are never ending and recurring means that the digits in them are repeating following a certain pattern. Such decimals can also be converted into rational numbers. E.g., 1.33333333…… = 4/3.
5. Irrational numbers: Non terminating non-recurring decimals are called irrational numbers. E.g., √2, √3, √5 + 7, , π etc. are all irrational numbers.
6. Real Numbers: The set of numbers which include both rational as well as irrational numbers is called set of real numbers. Real numbers are the ones which can be represented on the number line.
7. Prime Numbers: Numbers which have exactly two factors, one and the number itself, are called the prime numbers. The least prime number is 2 and 2 is the only prime number which is even. There are 25 prime numbers up to 100 and you should remember all these first 25 prime numbers.
Any prime number greater than 3, can be written as 6n + 1 or 6n – 1, where n is a natural number.
E.g., 23 is a prime number and 23 = 6 × 4 – 1. Similarly, 79 is a prime number which can be written as 6 × 13 + 1.
Note: Prime numbers can be written as 6n + 1 or 6n – 1 but every number which can be written as 6n + 1 or 6n – 1 need not to be a prime. E.g., 25 = 6 × 4 + 1, but 25 is not a prime number.
Any prime number divided by 6 leaves remainder 1 or 5.
Let us now discuss some examples:
1. The product of two integers is 28, what could be their minimum possible sum?
Sol: Let the integers are ‘a’ and ‘b’.
Given that a × b = 28
Now as, a and b are integers and minimum sum is asked, so we can take the values of a and b as negative.
Now, in such questions try to take one number which is equal to the product but with negative sign and adjust the other number accordingly. So, we can take a = – 28 and b = – 1 so that (- 28) × (- 1) = 28
The sum in this case is – 28 – 1 = – 29.
2. The product of two natural numbers is 28, what could be their minimum possible sum?
Sol: Let the natural numbers are ‘a’ and ‘b’.
Given that a × b = 28
Different possibilities are 1 × 28, 2 × 14, 4 × 7. The sum in these cases is 1 + 28 = 29, 2 + 14 = 16 and 4 + 7 = 11.
The least sum is 11.
In such questions where natural numbers are asked, take the numbers which are as close as possible and satisfy the given product also.
3. The product of two numbers is 28, what could be their minimum possible sum?
Sol: Now in this question, the specific type of the numbers is not given to us. The statement only mentions about the numbers which means real numbers. Here we can take the rational numbers also.
One case is (– 56) × (- 1/2) = 28. The sum in this case is -56.5
But if we take the numbers as (-5600000) (-1/200000) = 28 and sum in this case is very small.
We can take even smaller numbers and get the product as 28.
Therefore, here we cannot find the minimum sum.
Now in the above three questions, observe that the question was same but the only difference was the type of numbers used. As we changed the type of numbers, the answers changed for the same question.
4. The sum of three natural numbers is 18. What could be their maximum product?
Sol: Let the three natural numbers are ‘a’, ‘b’ and ‘c’ such that a + b + c = 18.
Now in case of natural numbers, the product will be maximum if the difference in numbers is minimum. If we take the three numbers as 11, 6 and 1 then the product is 11 × 6 × 1 = 66.
If we take the numbers which are little bit closer as 10, 6 and 2, then the product is 10 × 6 × 2 = 120. You can see that in the second case the product is more as compare to the first case. So the product will keep on increasing if we take the numbers which are closer to each other and their sum is 18.
Therefore, the maximum sum will be obtained if the difference between the numbers is minimum say 0. In this case the numbers will be 6, 6, and 6 and their product will be 6 × 6 × 6 = 216.
Note that this rule is followed only in the case of natural numbers.
5. The sum of three integers is 18. What could be their maximum product?
Sol: Now in this question, we are given integers and not the natural numbers, therefore, the rule of the previous question is not applicable here. Here, we have choices of negative numbers also and by taking 2 negative numbers and one positive number, we can have a positive product.
Let us take the numbers as 200, – 100 and – 82 such that 200 + (- 100) + (- 82) = 18 and the product in this case will be 200 × (- 100) × (- 82) = 1640000.
On the other hand, if we take the numbers as 20000, – 19000 and – 982, even then the sum is 18 but the product will be much higher and still that will not be our answer as we can take even larger numbers where we can make the sum 18 and get much larger products. Hence, in this case the answer cannot be found.
From the above discussion, it is clear that while solving the questions of number system, the types of numbers cannot be ignored as by changing the type, the answers will be changed.
In the next article, we will learn the concept of Unit Digit.
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