Important Formulas Prime Time - Maths Olympiad Class 6 PDF Download
Common Multiples
Common multiples are numbers that are multiples of two or more given numbers. A common multiple of two numbers is divisible by each of those numbers without leaving a remainder.
Example: For the numbers 3 and 5, common multiples are 15, 30, 45, ... because each of these is divisible by both 3 and 5.
Factors
Factor - A number a is a factor of another number b if a divides b exactly (that is, b ÷ a leaves remainder 0).
Example: 4 is a factor of 12 because 12 ÷ 4 = 3.
Common Factors
A number that divides each number in a set exactly is called a common factor of those numbers.
Example: 4 is a common factor of 12 and 16 because 12 ÷ 4 = 3 and 16 ÷ 4 = 4.
Prime Numbers
A prime number is a number greater than 1 that can only be divided evenly by 1 and itself. For example, 7 is a prime number because the only numbers that divide 7 evenly are 1 and 7.
Composite Numbers
A composite number is a number that has more than two factors. This means it can be divided evenly by 1, itself, and at least one other number. For example, 12 is composite because it can be divided by 1, 2, 3, 4, 6, and 12.
Example: 12 is composite because its divisors are 1, 2, 3, 4, 6 and 12.
Prime Factorisation
Every integer greater than 1 can be expressed as a product of prime numbers.
Example: 84 expressed as a product of primes is:
84 = 2 × 2 × 3 × 7
When giving prime factorisation it is common to write repeated primes using powers; for 84 this is 84 = 2² × 3 × 7.
Try yourself: What is the prime factorization of 60?
Unique Factorisation
Fundamental Theorem of Arithmetic - Every integer greater than 1 has a unique prime factorisation, except for the order of the factors.
Co-prime Numbers:
Two numbers are co-prime if they have no common factors other than 1.Co-prime numbers need not both be prime; they simply must not share any prime factor.
Co-prime (relatively prime) - To determine if two numbers are co-prime, find their prime factorizations. If there are no common prime factors, they are co-prime.
Example :
- 80 = 2 × 2 × 2 × 2 × 5
- 63 = 3 × 3 × 7
There are no common prime factors between 80 and 63, so 80 and 63 are co-prime.
Factor Inclusion
If the prime factorisation of number A is contained within the prime factorisation of number B ,then A is a factor of B.
Divisibility Tests
Simple tests to check whether a number is divisible by another without performing long division.
- Divisible by 10: if the last digit is 0.
- Divisible by 5: if the last digit is 0 or 5.
- Divisible by 2: if the last digit is even (0, 2, 4, 6, 8).
- Divisible by 4: if the number formed by the last two digits is divisible by 4.
- Divisible by 8: if the number formed by the last three digits is divisible by 8.
- Divisible by 3: if the sum of the digits is divisible by 3.
- Divisible by 9: if the sum of the digits is divisible by 9.
- Divisible by 6: if the number is divisible by both 2 and 3.
- Divisible by 11: if the alternating sum of digits (sum of digits in odd positions minus sum of digits in even positions) if the result is 0 or a multiple of 11, the original number is divisible by 11.
Special Numbers
- Perfect square: a number that is the square of an integer. Example: 9 = 3 × 3.
- Perfect cube: a number that is the cube of an integer. Example: 27 = 3 × 3 × 3.
FAQs on Important Formulas: Prime Time
| 1. What is a prime number? |  |
| 2. How can you determine if a number is prime? | |
| 3. Can 1 be considered a prime number? | |
| 4. What is the importance of prime numbers in mathematics? | |
| 5. Are there an infinite number of prime numbers? | |
