Associate Content Manager
A number is a numeric value that can be represented as an integer, rational number, real number, or complex number. Numerical values are usually written using digits in the range of 0 to 9. The set of all numbers is Z, the integers are Q, and non-integers are R.
Types of Numbers used in Number System
[Click Here for Sample Questions]
Real Numbers
They are called real because they correspond to the basic building blocks of reality, also known as natural numbers. The Real numbers include the positive integers (1, 2, 3...) and their negative counterparts (-2, -3,...). It also all includes all rational numbers and irrational numbers.
Rational Numbers
A rational number can be defined as a number that can be expressed as a fraction. Any number in the form of p/q where q0 is a rational number.
Integers
An integer can be defined as a whole number that can be positive, negative, or zero. 0, -1, -35, 1, 56, 100 are all Integers.
Whole Numbers
A whole number can be defined as a positive number without a decimal or fractions. 0, 1, 2, 3, 10, 30, 100, and so on are whole numbers.
Natural Numbers
A natural number is any whole number greater than 1 and it is infinite.
Prime numbers
Prime numbers are a special subset of natural numbers. A prime number is a positive integer with exactly two distinct natural numbers that are both factors. Factors of prime numbers include 1 and the number itself. Essentially this means it can't be broken down any further: you cannot find other factors such as 2 or 3 or 4. The first few prime are 2, 3, 5, 7.
Properties of Prime Numbers
[Click Here for Sample Questions]
- The lowest prime number is 2.
- The lowest odd prime number is 3.
- For every prime number p ≥ 5, there is an integer q such that p = 2q + 1.
- If a and b are any two odd primes then a2 – b2 is composite. Also, a2+ b2 is composite.
- The remainder of the division of the square of a prime number p ≥ 5 divided by 12 is 1.
Prime Factorisation
[Click Here for Sample Questions]
Prime factorization is the process of breaking numbers down into prime factors. Breaking a number down into its prime factors means looking at its individual parts, and determining which of those parts are prime numbers. To find the prime factorization of 16, for example, you would need to divide the number by 2 four times. After each division 8, 4, 2, 1 will be the yield. The prime number in this set is 2. Therefore the prime factor of 16 is 2.
Let’s look at prime factorization with another example of the number 40. 40 has the following factors: 1, 2, 4, 5, 8, 10, 20, 40. Since Prime numbers are numbers that can not be divided by any number except for itself and 1, we see the prime factors of 40 are the following: 2 and 5.
Useful Shortcuts for Finding out Prime Numbers
Checking Prime Numbers
If you’ve ever been curious about how to check whether a number is prime or not using shortcuts, look no further!
Step I: Firstly, Calculate the square root of the number.
Step II: The square root is rounded up to the next higher number integer. Call this number z.
Step III: Divide the number N by all prime numbers under z
Step IV: If N is not divisible by any prime number below z, it means that N is a prime number.
For Example, To find if 229 is a prime number or not, we first have to find the square root, 229, which lies between 15 and 16. We consider 16 as it is the greater integer. Prime numbers less than or equal to 16 are 2, 3, 5, 7, 11, and 13. 229 is not divisible by any of these numbers. Hence, 229 is a prime number.
Finding a prime number
[Click Here for Sample Questions]
For Numbers below 49
To check if a number is a prime number below 49, you only have to divide it by 3 and you’ll find out!
For Numbers above 49 and below 121
To check if a number is a prime number above 49 and below 121, You will have to check if the number is divisible by 3 and 7. But, when you remember that, 77, 91, and 119 are not prime numbers, You can check the divisibility of all numbers under 121 by just dividing it by 3.
For example, 61 is a prime number because it is not divisible by 3 and it is neither 91 nor 119.
For Numbers above 121 and below 169
To check if a number is a prime number above 121 and below 169, You will have to check if the number is divisible by 3, 7, and 11. But when you remember that, 133, 143, and 161 are not prime numbers, you can check the divisibility of all numbers by just dividing them by 3.
For example, 149 is a prime number because it is not divisible by 3 and it is neither 133, 143 nor 161.
Practice Questions
Q1. Find the series of common multiples of 54 and 36 (b) 33, 45 and 60
Ans: The Least common multiple of 36 and 54 would be 108. The next common multiple will be 216, 324, and so on. Thus, the required series would be 108, 216,324, 432, 540, 648...
Q2. Find the series of common multiples of 33, 45 and 60
Ans: The LCM of 33, 45 and 60 = 60 x 3 x 11 =1980. Thus, the required series are 1980, 3960, 5940…
Q3. Find the greatest number, which will divide 215, 167, and 135 to leave the same remainder in each case.
Ans: 135 - 167 = 32,
while, 167 - 215 = 48.
Using the HCF of these differences, we can answer this question.
Hence, HCF of 32 and 48 = 16.
Q4. What is the smallest number that when increased by 6 is divisible by 36, 63, and 108?
Ans: The LCM of 36, 63, and 108 is 756. Hence, the required number is 750.
Previous Year Questions Asked in CAT
Q1. Find the largest number which can exactly divide 216, 252, 294.
Ans: To find the HCF of 216, 252, 294,
216 → 23 x 33
252 →22 x 7 x 32
294 → 21 x 31 x 72
HCF = 21 X 31 = 6
Q2. Three red lights flash every minute and five green lights flash every two minutes. How many times do both lights flash together in an hour if both start flashing at the same time?
Ans: Three red lights flash every minute and five green lights flash every two minutes. Therefore, red and green lights flash every 1/3 minute and 2/5 minute respectively. There is a 2-minute LCM between the two fractions. Hence they flash together every 2 min. So in an hour, they flash together 30 times.
Q3. The LCM of the two numbers is 936. If their HCF is 4 and one of the numbers is 72, the other is:
Ans: LCM x HCF = 936 x 4 = N1 x N2 → 936 x 4 = 72 x N2 → N2= 13 x 4 = 52.
Q4. Two alarm clocks ring their alarms at regular intervals of 50 seconds and 48 seconds. If they first beep together at noon, at what time will they beep again for the first time?
Ans: The first time the alarm clocks would ring together would be after a time that is equal to the LCM of 50 and 48. The LCM of 50 and 48 is 50 x 24 = 1200. Hence, the first time they would ring together after noon would be exactly 1200 seconds or 20
minutes later.
Q5. The smallest square number, which is exactly divisible by 2, 3, 4, – 9, 6, 18, 30, and 60, is
Ans: The LCM of the given numbers is 180. As a result, these numbers are all divisible by all multiples of 180. Checking the series 180, 360, 540, 720, and 900 we can see that 900 is the first perfect square on the list.
Q6. On Ashok Marg three consecutive traffic lights change after 36, 42, and 72 seconds, respectively. If the lights are first switched on at 9:00 A.M. sharp, How long will it take them to change at the same time?
Ans: The LCM of 36,42 and 72 is 504. After 8 minutes and 24 seconds, the lights will switch on simultaneously.
Q7. In a row of 44 apple trees, 66 banana trees, and 110 mango trees, the forester wants to plant 44 apple trees, 66 banana trees, and 110 mango trees (according to their number). Also, he wants to separate trees into distinct rows (e.g. only one type of tree per row). The number of rows (minimum) that are required are
Ans: 44/22 + 66/22 + 110/22 (Since 22 is the HCF)
Q8. A piece of timber 42 meters long, 49 meters long, and 63 meters long has to be divided into four planks of equal length. What will be the minimum possible number of planks?
Ans: Assuming that we divide each plank into a length equal to 43, 49, and 63, we obtain the smallest number of planks possible. These numbers have an HCF of 7-, so this is the size of each plank. Using this example, there would be a total of:
42/7 + 49/7 + 63/7 = 6 + 7 + 9 = 22 planks.
Q9. Find the greatest number, which will divide 215, 167, and 135 to leave the same remainder in each case.
Ans: 135 - 167 = 32,
while, 167 - 215 = 48.
Using the HCF of these differences, we can answer this question.
Hence, HCF of 32 and 48 = 16.
Q10. Among all the square numbers that are exactly divisible by 2, 3, 4, – 9, 6, 18, 30, and 60, one is the smallest
Ans: The LCM of the given numbers is 180. These numbers are all divisible by 180, so all multiples of 180 are also divisible by all of these numbers. The first perfect square on the list can be found by checking the series 180, 360, 540, and 720,900.
Comments