Playing with Numbers Summary Class 6 Maths Del

Factors

• A factor of a number is an exact divisor of that number.
• 1 is a factor of every number.
• Every number is a factor of itself.
• Every factor is less than or equal to the given number.
• Number of factors of a given number are finite.

Multiples

• A number is said to be a multiple of any of its factors.
• Every multiple of a number is greater than or equal to that number.
• The number of multiples of a given number is infinite.
• Every number is a multiple of itself.

Prime and Composite Numbers

• Prime numbers: Each of the numbers which have exactly two factors namely, 1 and the number itself, are called prime numbers, For example, 2, 3, 5, 7, 11, .... etc. 2 is the smallest prime number which is even. Every prime number except 2 is odd.
• Twin prime: Two consecutive odd prime numbers are known as twin prime. For example, 3 and 5.
• Prime triplet: A set of three consecutive prime numbers, differing by 2, is called a prime triplet. The only prime triplet is (3, 5, 7).
• Co-primes: Two numbers are said to be co-prime if they do not have a common factor other than 1. For example, 2 and 3. Any two prime numbers are always co-prime but converse is not always true.
• Composite numbers: The numbers having more than two factors are known as composite numbers.
  For example, 4, 6, 8, 9, 10, 12, 14, .... etc. and 1 is neither prime nor composite because it has a single factor.
• Even number: All multiples of 2 are called even number.
• Odd number: Numbers which are not multiples of 2 are called odd numbers.

Tests for Divisibility of Numbers

A number is divisible by 2, if its unit digit are 0, 2, 4, 6 or 8.

• A number is divisible by 3, if the sum of its digits is divisible by 3.
• A number is divisible by 4, if the number formed by last two digits is divisible by 4.
• A number is divisible by 5, if its unit digit is 0 or 5.
• A number is divisible by 6, if the number is divisible by 2 and 3.
• A number is divisible by 8, if the number formed by last three digits is divisible by 8.
• A number is divisible by 9, if the sum of its digits is divisible by 9.
• A number is divisible by 11, if the difference of the sum of its digits in odd places and the sum of its digits in even places is either 0 or a multiple of 11.

Common Factors and Common Multiples

• A factor which is common in two or more numbers is called common factor.
• A multiple which is common to two or more numbers is called common multiple.
• Co-prime are those number which have 1$1$ as common factors.

Factor Tree

When we go on factorizing a number till we reach its ultimate factorisation and write the process as follows, we get the shape of a tree called Factor tree.
For example: Let us factorize 90. Let us see below: 

Thus. 90 = 2 × 3 × 3 × 5. 

Prime Factorisation

The method of representing a number as a product of its factors is called prime factorization.

Highest Common Factor

HCF of two or more numbers is the highest or greatest factor among all the common factors.

Lowest Common Multiple

• LCM of two or more numbers is the lowest multiple among all the common multiples
• Simply, a product of the prime factors of two or more numbers is called LCM, counted by the maximum number of times they occur in any of the numbers.

Some More Divisibility Rules

• 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 is also divisible by that number.
• If two given numbers are divisible by a number, then their difference is also divisible by that number.