Playing with Numbers
A factor of a number is an exact divisor of that number.
Properties of factors:
Factors of a number are always less than or equal to the number itself and are finite.
Every number is a factor of itself.
1 is a factor of every number.
A number for which sum of all its factors is equal to twice the number is called a perfect number.
A multiple of a number is a number obtained by multiplying it by a natural number.
Properties of Multiples:
Every multiple of a number is greater than or equal to that number and are infinite.
Every number is a multiple of itself.
The numbers other than 1 whose only factors are 1 and the number itself are called prime numbers. Numbers having more than two factors are called composite numbers.
1 is neither prime nor composite.
2 is the smallest prime number which is even.
Tests for divisibility of numbers:
A number is divisible by 2 if it has any of the digits 0, 2, 4, 6, or 8 in its ones place.
A number is divisible by 3 if the sum of its digits is a multiple of 3.
A number with three or more digits is divisible by 4 if the number formed by its last two digits (i.e., digits at ones and tens places) is a multiple of 4.
A number is divisible by 5 if it has either 0 or 5 in its ones place.
A number of 4 or more digits is divisible by 8 if the number formed by the last three digits is divisible by 8.
A number is divisible by 9 if the sum of its digits is a multiple of 9.
A number is divisible by 10 if it has 0 in its ones place.
A number is divisible by 11 if the difference between the sum of digits at odd places (from the right) and the sum of digits at even places (from the right) is either 0 or a multiple of 11.
Two numbers having only 1 as a common factor are called co-prime numbers.
If a number is divisible by another number then it is divisible by each of the factors of that number.
If a number is divisible by two co-prime numbers then it is divisible by their product also.
If two given numbers are divisible by a number, then their sum and difference are also divisible by that number.
When a number is expressed as a product of its factors, the prime factorization of the number has been done. The Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of two or more given numbers is the highest (or greatest) of their common factors. The Lowest Common Multiple (LCM) of two or more given numbers is the lowest (or smallest or least) of their common multiples.
KEYWORDS: Sieve of Eratosthenes, Factor tree