Number System-1(types of number and divisibility Rule)

Quantitative Aptitude ( CCS) Number System

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NUMBER SYSTEM

Types of Numbers

Natural Numbers: The numbers 1,2,3,4.... are called natural numbers or positive integers.
Whole Numbers: The numbers 0,1,2,3.... are called whole numbers. Whole numbers include “0”.
Integers: The numbers .... -3, -2, -1, 0, 1, 2, 3,.... are called integers. You will see questions on integers in almost all the exams where you see number system aptitude questions.
Negative Integers: The numbers -1, -2, -3, ... are called negative integers.
Positive Fractions: The numbers(2/3) ,(4/5) ,(7/8) ... are called positive fractions.
Negative Fractions: The numbers -(6/8) ,-(7/19) , -(12/17) ... are called negative fractions.
Rational Numbers: Any number which is a positive or negative integer or fraction, or zero is called a rational number. A rational number is one which can be expressed in the following format ⇒(a/b) , where b ≠ 0 and a & b are positive or negative integers.
Irrational Numbers: An infinite non-recurring decimal number is known as an irrational number. These numbers cannot be expressed in the form of a proper fraction a/b where b ≠ 0. e.g.√2 , √5 , Π, etc.
Surds: Any root of a number, which cannot be exactly found is called a surd. Essentially, all surds are irrational numbers. e.g. √2 , √5 etc
Prime Numbers: Those numbers, which are divisible only by themselves and 1, are called prime numbers. In other words, a number, which has only two factors, 1 and itself, is called a prime number. e.g. 2, 3, 5, 7, etc.

Note:

2 is the only even prime number.
There are 25 prime numbers upto 100. These are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89 & 97. These should be learnt by heart.

Co-Prime Numbers: Two numbers are considered to be prime to each other if their HCF is 1. e.g. 5 and 24 are prime to each other. In other words, 5 and 24 are co-prime.
To check whether a number is prime, e.g. 79, we do not need to check all the factors below 79. The square of 8 is 64 & that of 9 is 81. Therefore, check if any of the prime numbers less than 9 is a factor of 79. The prime numbers 2, 3, 5, 7 are not the factors of 79. So, 79 is a prime number

Composite Number: A number, which has factors other than itself and 1, is called a composite number. e.g. 9, 16, 25....
Note: 1 is neither a composite number nor a prime number.

Illustration: Which of the following is a prime number?
1. 31

2. 91

3. 87

4. 57

Sol: Here 91 is divisible by 7, so 91 is not a prime number. 87 is divisible by 3. So it is also not a prime number. 57 is not a prime number as it is divisible by 3. 31 has only two factors 1 and 31, so it is a prime number.

Consecutive Numbers: Numbers arranged in increasing order and differing by 1 are called consecutive numbers. e.g. 4, 5, 6, 7 etc.

Real Numbers: The natural numbers, integers, whole numbers, rational numbers and irrational numbers constitute the set of real numbers. Every real number can be represented by a point on a number line.

Perfect Numbers: If the sum of all the factors of a number excluding the number itself happens to be equal to the number, then the number is called as perfect number. 6 is the first perfect number. The factors of 6 are 1, 2, 3 & 6. Leaving 6 the sum of other factors of 6 are equal to 6. The next three perfect numbers after 6 are: 28,496 and 8128.

Complex Numbers: Complex numbers have a real and an imaginary component; e.g. (√-2 - 4), (2 +√-3 ), etc. Square root of any negative number is an imaginary number - e.g.√-2 ,√-3 . The square root of a negative number does not exist in the real sense. Such numbers are called imaginary numbers.

Fibonacci Numbers: The numbers, which follow the following series are known as Fibonacci numbers. E.g. 1,1,2,3,5,8,13,21..... The series is obtained by adding the sum of the preceding two numbers. In general for a Fibonacci number X, X_i+2 = X_i+1 + X_i.

Even numbers: Even numbers end in 0, 2, 4, 6, 8 and 0 .

Odd numbers: An odd number is a number that cannot be divided into two equal groups.
Odd numbers end in 1, 3, 5, 7, 9.

Operations on Odd & Even Numbers

odd no + odd no = even number
odd no + even no = odd number
even no + even no = even no
odd no - odd no = even number
odd no - even no = odd number
even no -even no = even no

Multiplication of two odd numbers will result in an odd number. E.g. 5 × 7 = 35.

Multiplication of two even numbers results in an even number. E.g. 4 × 6 = 24.

An even number raised to an odd or an even power is always even.

Divisibility Rule

Divisor	Divisibility condition	Examples
1	No special condition. Any integer is divisible by 1.	2 is divisible by 1.
2	The last digit is even (0, 2, 4, 6, or 8).	1294: 4 is even.
3	Sum the digits. The result must be divisible by 3.	405 → 4 + 0 + 5 = 9 and 636 → 6 + 3 + 6 = 15 which both are clearly divisible by 3. 16,499,205,854,376 → 1+6+4+9+9+2+0+5+8+5+4+3+7+6 sums to 69 → 6 + 9 = 15 → 1 + 5 = 6, which is clearly divisible by 3.
3	Subtract the quantity of the digits 2, 5, and 8 in the number from the quantity of the digits 1, 4, and 7 in the number. The result must be divisible by 3.	Using the example above: 16,499,205,854,376 has four of the digits 1, 4 and 7 and four of the digits 2, 5 and 8; ∴ Since 4 − 4 = 0 is a multiple of 3, the number 16,499,205,854,376 is divisible by 3.
4	The last two digits form a number that is divisible by 4.	40,832: 32 is divisible by 4.
	If the tens digit is even, the ones digit must be 0, 4, or 8. If the tens digit is odd, the ones digit must be 2 or 6.	40,832: 3 is odd, and the last digit is 2.
	Twice the tens digit, plus the ones digit is divisible by 4.	40832: 2 × 3 + 2 = 8, which is divisible by 4.
5	The last digit is 0 or 5	495: the last digit is 5.
6	It is divisible by 2 and by 3.	1458: 1 + 4 + 5 + 8 = 18, so it is divisible by 3 and the last digit is even, hence the number is divisible by 6.
7	Forming an alternating sum of blocks of three from right to left gives a multiple of 7	1,369,851: 851 − 369 + 1 = 483 = 7 × 69
	Adding 5 times the last digit to the rest gives a multiple of 7. (Works because 49 is divisible by 7.)	483: 48 + (3 × 5) = 63 = 7 × 9.
	Subtracting 2 times the last digit from the rest gives a multiple of 7. (Works because 21 is divisible by 7.)	483: 48 − (3 × 2) = 42 = 7 × 6.
	Subtracting 9 times the last digit from the rest gives a multiple of 7.	483: 48 − (3 × 9) = 21 = 7 × 3.
	Adding 3 times the first digit to the next and then writing the rest gives a multiple of 7. (This works because 10a + b − 7a = 3a + b; the last number has the same remainder as 10a + b.)	483: 4×3 + 8 = 20, 203: 2×3 + 0 = 6, 63: 6×3 + 3 = 21.
	Adding the last two digits to twice the rest gives a multiple of 7. (Works because 98 is divisible by 7.)	483,595: 95 + (2 × 4835) = 9765: 65 + (2 × 97) = 259: 59 + (2 × 2) = 63.
	Multiply each digit (from right to left) by the digit in the corresponding position in this pattern (from left to right): 1, 3, 2, -1, -3, -2 (repeating for digits beyond the hundred-thousands place). Adding the results gives a multiple of 7.	483,595: (4 × (-2)) + (8 × (-3)) + (3 × (-1)) + (5 × 2) + (9 × 3) + (5 × 1) = 7.
	Compute the remainder of each digit pair (from right to left) when divided by 7. Multiply the rightmost remainder by 1, the next to the left by 2 and the next by 4, repeating the pattern for digit pairs beyond the hundred-thousands place. Adding the results gives a multiple of 7.	194,536: 19\|45\|36 ; (5x4) + (3x2) + (1x1) = 27, so it is not divisible by 7 204,540: 20\|45\|40 ; (6x4) + (3x2) + (5x1) = 35, so it is divisible by 7
8	If the hundreds digit is even, the number formed by the last two digits must be divisible by 8.	624: 24.
	If the hundreds digit is odd, the number obtained by the last two digits plus 4 must be divisible by 8.	352: 52 + 4 = 56.
	Add the last digit to twice the rest. The result must be divisible by 8.	56: (5 × 2) + 6 = 16.
	The last three digits are divisible by 8.	34,152: Examine divisibility of just 152: 19 × 8
	Add four times the hundreds digit to twice the tens digit to the ones digit. The result must be divisible by 8.	34,152: 4 × 1 + 5 × 2 + 2 = 16
9	Sum the digits. The result must be divisible by 9.	2880: 2 + 8 + 8 + 0 = 18: 1 + 8 = 9.
10	The ones digit is 0.	130: the ones digit is 0.
11	Form the alternating sum of the digits. The result must be divisible by 11.	918,082: 9 − 1 + 8 − 0 + 8 − 2 = 22 = 2 × 11.
	Add the digits in blocks of two from right to left. The result must be divisible by 11.	627: 6 + 27 = 33 = 3 × 11.
	Subtract the last digit from the rest. The result must be divisible by 11.	627: 62 − 7 = 55 = 5 × 11.
	Add the last digit to the hundreds place (add 10 times the last digit to the rest). The result must be divisible by 11.	627: 62 + 70 = 132: 13 + 20 = 33 = 3 × 11.
	If the number of digits is even, add the first and subtract the last digit from the rest. The result must be divisible by 11.	918,082: the number of digits is even (6) → 1808 + 9 − 2 = 1815: 81 + 1 − 5 = 77 = 7 × 11
	If the number of digits is odd, subtract the first and last digit from the rest. The result must be divisible by 11.	14,179: the number of digits is odd (5) → 417 − 1 − 9 = 407 = 37 × 11
12	It is divisible by 3 and by 4.	324: it is divisible by 3 and by 4.
12	Subtract the last digit from twice the rest. The result must be divisible by 12.	324: 32 × 2 − 4 = 60 = 5 × 12.
13	Form the alternating sum of blocks of three from right to left. The result must be divisible by 13.	2,911,272: 272 - 911 + 2 = -637
	Add 4 times the last digit to the rest. The result must be divisible by 13.	637: 63 + 7 × 4 = 91, 9 + 1 × 4 = 13.
	Subtract the last two digits from four times the rest. The result must be divisible by 13.	923: 9 × 4 - 23 = 13.
	Subtract 9 times the last digit from the rest. The result must be divisible by 13.	637: 63 - 7 × 9 = 0.
14	It is divisible by 2 and by 7	224: it is divisible by 2 and by 7.
14	Add the last two digits to twice the rest. The result must be divisible by 14.	364: 3 × 2 + 64 = 70. 1764: 17 × 2 + 64 = 98.
15	It is divisible by 3 and by 5.	390: it is divisible by 3 and by 5.
16	If the thousands digit is even, the number formed by the last three digits must be divisible by 16.	254,176: 176.
	If the thousands digit is odd, the number formed by the last three digits plus 8 must be divisible by 16.	3408: 408 + 8 = 416.
	Add the last two digits to four times the rest. The result must be divisible by 16.	176: 1 × 4 + 76 = 80. 1168: 11 × 4 + 68 = 112.
	The last four digits must be divisible by 16.	157,648: 7,648 = 478 × 16.
17	Subtract 5 times the last digit from the rest.	221: 22 − 1 × 5 = 17.
	Subtract the last two digits from two times the rest.	4,675: 46 × 2 - 75 = 17.
	Add 9 times the last digit to 5 times the rest. Drop trailing zeroes.	4,675: 467 × 5 + 5 × 9 = 2380; 238: 23 × 5 + 8 × 9 = 187.
18	It is divisible by 2 and by 9	342: it is divisible by 2 and by 9.
19	Add twice the last digit to the rest.	437: 43 + 7 × 2 = 57.
19	Add 4 times the last two digits to the rest.	6935: 69 + 35 × 4 = 209.
20	It is divisible by 10, and the tens digit is even.	360: is divisible by 10, and 6 is even.
20	The number formed by the last two digits is divisible by 20	480: 80 is divisible by 20.