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# 7. Composite Numbers: All counting number which is not prime, are called a composite number.

A composite has more than two factors.

8. Co-prime: Two numbers whose HCF is 1 are called co-prime numbers.

Ex. (2, 3), (8, 9) are pairs of co-prime.

9. Rational Numbers: Numbers that can be expressed in the form of p/q, whose p and q are integers and q≠0, are called rational numbers.

Ex. 1/8, 2/5, 0, 6, 9/8 etc.

10. Irrational Numbers:- Numbers which when expressed in decimal would be in the non-terminating and non-repeating forms are called irrational numbers.

Ex. √2, √3, √5, √7, Π,

POSITIVE AND NEGATIVE NUMBERS

A positive number is a real number that is greater than zero. A negative number is a real number that is smaller than zero.

Zero is not positive, nor negative.

Multiplication:

• positive * positive = positive
• positive * negative = negative
• negative * negative = positive

Division:

• positive / positive = positive
• positive / negative = negative
• negative / negative = positive

## Prime Numbers

A Prime number is a natural number with exactly two distinct natural number divisors: 1 and itself. Otherwise, a number is called a composite number. Therefore, 1 is not a prime, since it only has one divisor, namely 1. A number n>1 is prime if it cannot be written as a product of two factors a and b, both of which are greater than 1: n = ab.

• The first twenty-six prime numbers 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, 101

• Note: only positive numbers can be primes.

• There are infinitely many prime numbers.

• The only even prime number is 2, since any larger even number is divisible by 2. Also 2 is the smallest prime.

• All prime numbers except 2 and 5 end in 1, 3, 7 or 9, since numbers ending in 0, 2, 4, 6 or 8 are multiples of 2, and numbers ending in 0 or 5 are multiples of 5. Similarly, all prime numbers above 3 are of the form 6n-1 or  6n+1 , because all other numbers are divisible by 2 or 3.

• Any nonzero natural number n can be factored into primes, written as a product of primes or powers of primes. Moreover, this factorization is unique except for a possible reordering of the factors.

• Prime factorization: every positive integer greater than 1 can be written as a product of one or more prime integers in a way that is unique. For instance integer n with three unique prime factors a, b and c can be expressed as n = ap*bq*cr, where p, q, and r are powers of a, b, and c, respectively, and are >1.

Example: 4200 = 23*3*52*7 .

Verifying the primality (checking whether the number is a prime) of a given number n can be done by trial division, that is to say, dividing by all integer numbers smaller than √n, thereby checking whether n is a multiple of m<√n.

Example: Verifying the primality of 161:√161 is little less than 13, from integers from 2 to 13, 161 is divisible by 7, hence 161 is not prime.

• If n is a positive integer greater than 1, then there is always a prime number P with n<p<2n.

Factor

A divisor of an integer n, also called a factor of n, is an integer that evenly divides n without leaving a remainder. In general, it is said m is a factor of n, for non-zero integers m and n, if there exists an integer k such that n = km.

• 1 (and -1) are divisors of every integer.

• Every integer is a divisor of itself.

• Every integer is a divisor of 0, except, by convention, 0 itself.

• Numbers divisible by 2 are called even and numbers not divisible by 2 are called odd.

• A positive divisor of n which is different from n is called a proper divisor.

• An integer n > 1 whose only proper divisor is 1 is called a prime number. Equivalently, one would say that a prime number is one which has exactly two factors: 1 and itself.

• Any positive divisor of n is a product of prime divisors of n raised to some power.

• If a number equals the sum of its proper divisors, it is said to be a perfect number. Example: The proper divisors of 6 are 1, 2, and 3: 1+2+3=6, hence 6 is a perfect number.

There are some elementary rules:

• If a is a factor of b and a is a factor of c, then a is a factor of (b+c)

. In fact, a is a factor of (mb+nc) for all integers m and n.

• If a is a factor of b and b is a factor of c, then a is a factor of c.

• If a is a factor of b and b is a factor of a, then a=b or b=a.

• If a is a factor of bc, and gcd(a,b)=1, then a is a factor of c.

• If p is a prime number and p is a factor of ab then p is a factor of a or p is a factor of b.

Finding the Number of Factors of an Integer

First make prime factorization of an integer n=ap*bq*cr, where a, b, and c are prime factors of n and P, q, and r are their powers.

The number of factors of n will be expressed by the formula (p+1)(q+1)(r+1). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: 450=21*32*52

Total number of factors of 450 including 1 and 450 itself is factors.(1+1)*(2+1)*(2+1)= 18 factors

Finding the Sum of the Factors of an Integer

First make prime factorization of an integer n = ap*bq*cr, where a, b, and c are prime factors of and p, q, and r are their powers.

The sum of factors of n will be expressed by the formula: Example: Finding the sum of all factors of 450:450=21*32*55

The sum of all factors of 450 is Greatest Common Factor (Divisor) – GCF (GCD)

The greatest common divisor (GCD), also known as the greatest common factor (GCF), or highest common factor (HCF), of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.

To find the GCF, you will need to do prime-factorization. Then, multiply the common factors (pick the lowest power of the common factors).

• Every common divisor of a and b is a divisor of GCD (a, b)

• a*b=GCD(a, b)*lcm(a, b)

Perfect Square

A perfect square is an integer that can be written as the square of some other integer. For example, 16=4^2 is a perfect square.

There are some tips about the perfect square:

• The number of distinct factors of a perfect square is ALWAYS ODD.

• The sum of distinct factors of a perfect square is ALWAYS ODD.

• A perfect square ALWAYS has an ODD number of Odd-factors and EVEN number of Even-factors.

• Perfect square always has an even number of powers of prime factors.

Divisibility Rules

2 – If the last digit is even, the number is divisible by 2.

3 – If the sum of the digits is divisible by 3, the number is also.

4 – If the last two digits form a number divisible by 4, the number is also.

5 – If the last digit is a 5 or a 0, the number is divisible by 5.

6 – If the number is divisible by both 3 and 2, it is also divisible by 6.

7 – Take the last digit, double it, and subtract it from the rest of the number, if the answer is divisible by 7 (including 0), then the number is divisible by 7.

8 – If the last three digits of a number are divisible by 8, then so is the whole number.

9 – If the sum of the digits is divisible by 9, so is the number.

10 – If the number ends in 0, it is divisible by 10.

11 – If you sum every second digit and then subtract all other digits and the answer is: 0, or is divisible by 11, then the number is divisible by 11.

Example: to see whether 9,488,699 is divisible by 11, sum every second digit: 4+8+9=21, then subtract the sum of other digits: 21-(9+8+6+9)=-11, -11 is divisible by 11, hence 9,488,699 is divisible by 11.

12 – If the number is divisible by both 3 and 4, it is also divisible by 12.

25 – Numbers ending with 00, 25, 50, or 75 represent numbers divisible by 25

Factorials

Factorial of a positive integer n, denoted by n!, is the product of all positive integers less than or equal to n. For instance 5! = 1*2*3*4*5

• Note: 0!=1.

• Note: The factorial of negative numbers is undefined.

Trailing zeros: Trailing zeros are a sequence of 0’s in the decimal representation (or more generally, in any positional representation) of a number, after which no other digits follow.

125000 has 3 trailing zeros;

The number of trailing zeros in the decimal representation of n!, the factorial of a non-negative integer n, can be determined with this formula:

n/5+n/52+n/53………+n/5k,, where k must be chosen such that 5k<1.

Finding the number of powers of a prime number P, in the n!.

The formula is: n/p+n/p2+n/p3……..till px<n.

What is the power of 2 in 25!?

25/2+25/4+25/16 = 12+6+3+1 = 22