Arithmetic – Factorization Part1 – Prime Numbers

Prime Number

• A number is a prime number if it is divided by one and itself.

• E.g: 2, 3, 5, 7, 11, 13, 17, 19

• 1 is not a prime number.

• 2 is the only even prime number.

• How to check whether is number is prime or not ?

• Even numbers are not prime [except 2. Look for 2, 4, 6, 8 in the units place]

• Check whether it is divisible by a prime 3, 5, 7.

• Check for the remainders and last digits.

Divisibility and Factors

• Given two numbers, if the largest is divided by smallest number with

remainder 0.

• E.g: 39, 13 -> 39/13 = 3

• 39 is divided by 13. Hence 13 is the divisor of 39

• 13 is also called as Factor of 39.

• Sum of the divisors of 45?

• Sol:

Number of factors

• Factors of a given a number X:

• If X is divided any number y with a remainder 0, then y is called factor of X.

• E.g: 45/5 = 9 with remainder 0. Hence 5 is called as factor of 45.

• Number of factors/divisors of a given number:

• Select the numbers from 1 to X exclusive. [By default 1 and X are factors of X]

• Divide X by each number Y and check for remainder 0.

• If remainder 0, then y is a factor of X.

• Easy way to find the number of factors ?

Prime Factors

• Factors which are prime numbers too.

• E.g: Given number is 36, number of prime numbers are.

• 2, 3.

• How to find the prime factors?

• Given number is 36. Split the number by writing it as product of its factors.

• 36 = 9 * 4

= 3 * 3 * 2 * 2

• Cannot be split further as 2 and 3 prime numbers.

• Hence the remaining numbers in the product are prime numbers.

• Unique Factorization Theorem

Unique Factorization Theorem

• Also called as Fundamental Theorem of arithmetic

• Any given integer is prime by itself or a product of prime numbers

[factors].

• The product is unique upto the order of the factors.

• E.g: Find the prime factor product for 900

• 900 = 9 * 100

= 3 * 3 * 5 * 5 * 2 * 2 [2 ,3 ,5 are prime numbers. Hence this is the final

product in the order of prime factors]

Problems:

• Find the number of distinct prime factors of 288?

• Sol:

• Find the sum of all the factors of 56?

• Sol:

Problems:

• Sum of 4 unique prime factors is equal to 10. What is the least

number for which these numbers are prime factors?

• Sol: