Classes of Numbers
Natural numbers are 1, 2, 3 and so on, while whole numbers include zero. A prime number has no factors other than 1 and itself. Two is the only even prime, and there are 25 primes below 100. A composite number has more than two factors, and 1 is neither prime nor composite. Two numbers are co-prime if their highest common factor is 1.
Divisibility Rules
A number is divisible by 2 if its last digit is even. It is divisible by 3 or 9 if its digit sum is divisible by 3 or 9. It is divisible by 4 if its last two digits are, and by 8 if its last three digits are. It is divisible by 5 if it ends in 0 or 5. For 11, the difference between the sum of digits in odd places and the sum in even places must be zero or a multiple of 11.
HCF and LCM
The product of two numbers equals their HCF times their LCM. The HCF of fractions is the HCF of the numerators over the LCM of the denominators. The LCM of fractions is the LCM of the numerators over the HCF of the denominators. If a number leaves the same remainder r when divided by a, b and c, it equals the LCM times k plus r. If the shortfall from each divisor is the same, subtract that shortfall from a multiple of the LCM. The largest number that divides a and b leaving remainder r is the HCF of a minus r and b minus r.
Number of Factors
If N equals p to the a times q to the b, the number of factors is a plus 1 times b plus 1.
Sums
The sum of the first n natural numbers is n times n plus 1 over 2. The sum of the first n odd numbers is n squared. The sum of the first n even numbers is n times n plus 1. The sum of squares is n times n plus 1 times 2n plus 1 over 6. The sum of cubes is the square of the sum of the first n numbers.
Shortcuts
Use the division method for LCM and the Euclidean method for HCF. The product of any three consecutive numbers is divisible by 6. The last digit of a perfect square is never 2, 3, 7 or 8.
Worked Examples
Example 1: Find the LCM and HCF of 12, 18 and 24.
Solution: The LCM is 72 and the HCF is 6.
Example 2: The HCF of two numbers is 6, the LCM is 72, and one number is 18. Find the other.
Solution: The product is 432, and 432 divided by 18 equals 24.
Example 3: Find the number of factors of 360.
Solution: 360 equals 2 cubed times 3 squared times 5, so the count is 4 times 3 times 2, which is 24.
Quick Check
Question 1: What is the only even prime?
Answer: Two.
Question 2: How many primes are below 100?
Answer: Twenty-five.
Question 3: What is the sum of the first n odd numbers?
Answer: n squared.