NUMBERS ( Important Fact And Formula )

Key Points on Number Types:

1. Natural Numbers (N): These are counting numbers starting from 1, such as 1, 2, 3, 4, etc. They are used to count objects. The digit ‘0’ is considered insignificant, while all other digits are significant.
2. Whole Numbers (W): When ‘0’ is included with the natural numbers, we get the whole numbers, i.e., 0, 1, 2, 3, 4, etc. Whole numbers are non-negative and contain no fractions or decimals.
3. Integers (Z): Integers are whole numbers that include both positive and negative numbers, as well as zero, e.g., -5, -4, -3, 0, 1, 2, etc. They do not include fractions or decimal values.
4. Prime Numbers: A prime number is a whole number greater than 1 that has no divisors other than 1 and itself. Examples include 2, 3, 5, 7, and 11.
5. Co-prime Numbers: Two natural numbers, $p$ and $q$, are co-prime if their highest common factor (H.C.F.) is 1. Examples: (2, 3), (4, 5), (7, 9), (11, 9). Co-primes don’t necessarily have to be primes themselves, but they must share no common divisors other than 1.
6. Composite Numbers: A composite number is a whole number greater than 1 that isn’t a prime number. It can be divided by numbers other than 1 and itself. Examples include 4, 6, 8, 9, 10, and 12.
7. Even Numbers: An integer divisible by 2 is called an even number. Examples include 2, 4, 6, and 8.
8. Odd Numbers: An integer that isn’t divisible by 2 is called an odd number. Examples include 1, 3, 5, 7, and 9.
9. Consecutive Numbers: A series of numbers in which each successive number is 1 greater than the previous one. Examples include sequences like 1, 2, 3, 4, and 5.
10. Rational Numbers: Numbers that can be expressed in the form p/q where p and q are integers and q≠0 are called rational numbers. Examples include 22/7 , 5/7 , -143/15 .
11. Irrational numbers: The numbers which when written in decimal form do not terminate and repeat are known as irrational numbers. These numbers cannot be expressed as a ratio of integers or as a fraction, e.g.√2,√3,√5, π, etc.12) Real Numbers: The numbers which can be found on the number line and include both rational and irrational numbers are known as real numbers, e.g. -1.5,√2,0,1,2,3,π.Almost any number which you can imagine is a real number.

    13) Face value: It is the actual value of a digit. It remains definite and does not change with the digit’s place, e.g. in the numbers 435 and 454, the digit 5 has a face value of 5.

    14) Place value: The place value of a digit depends on its place or position in the number. It is the product of its place and face value. Each place in a number has a value of 10 times the place to its right, e.g. in a number 567, the digit 7 is in the ones place, digit 6 is in tens place and digit 5 is in hundreds place and the place value of 7 is 7*10⁰ = 7, the place value of 6 is 6* 10¹ = 60 and place value of 5 is 5 * 10² = 500.

    15) Unit digit: The unit digit is the ones place digit of a number.

    Rule for 0, 1, 5, and 6: The integers that end in digits 0, 1, 5, and 6 have the same unit digit, e.g. 0,1,5,6 respectively, irrespective of the positive integer exponent.

    For example: The unit digit of 156⁴ = 6
    Similarly, the unit digit of 131^1783663797 =1

Rule for integer that end in 4:

The unit digit of 4¹= 4
The unit digit of 4² = 16
The unit digit of 4³ = 64
The unit digit of 4⁴ = 256

It shows if the power of 4 is even, the unit digit is 6, and if the power is an odd number, the unit digit is 4.

Rule for 9:

9¹ = 9
9² = 81
9³ = 729
9⁴= 6561

It shows if the power of 9 is even, the unit digit is 1, and if the power is an odd number, the unit digit is 9.

Rule for 2, 3, 7, and 8: These numbers have a power cycle of 4 different numbers as show below:

2¹ =2
2² =4
2³ = 8
2⁴ = 16
2⁵ = 32
2⁶ = 64
2⁷ = 128
2⁸ = 256

It follows a pattern: 2,4,8,6, 2,4,8,6 and so on.

So, the possible unit digit of 2 has 4 different numbers 2, 4, 8, and 6.

Question Pattern

• What is the difference in the place value of 5 in the numeral 754853?

1. 49500
2. 49950
3. 45000
4. 49940

• What should be added to 1459 so that it is exactly divisible by 12?

1. 4
2. 3
3. 5
4. 6

•  If the number 467X4 is divisible by 9, find the value of the digit marked as X.

1. 4
2. 5
3. 6
4. 7

• 7X2 is a three digit number and X is the missing digit. If the number is divisible by 6, the missing digit is

1. 4
2. 3
3. 2
4. 5

•  What smallest number should be subtracted from 9805 so that it is divisible by 8?

1. 3
2. 4
3. 5
4. 7