# Playing With Numbers

#### Prime Factorisation

Prime Factorisation. It is the process of finding the (prime) factors, which are multiplied together to get the original number (order in which the factors appear does not matter).

For example, we can write

• 24 = 2 × 2 × 2 × 3
• 24 = 2 × 2 × 3 × 2
• 24 = 2 × 3 × 2 × 2
• 24 = 3 × 2 × 2 × 2

and the same prime factors (there are three 2’s and one 3) occur in all the cases.

##### More divisibility rules
• Product of two consecutive whole numbers is divisible by 2
• 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.

Example. Find the prime factorisation of 324.

Prime factorization of 324

= 2 × 2 × 3 × 3 × 3 × 3.

Example. Find the prime factorisation of 400.

Prime factorization of 400

= 2 × 2 × 2 × 2 × 5 × 5.

Example. Find the prime factorisation of 20570.

Prime factorization of 20570 = 5 × 2 × 11 × 11 × 17.

Example. Find the prime factorisation of 58500.

Prime factorization of 58500 = 5 × 5 × 5 × 3 × 3 × 2 × 2 × 13.

Example. Find the prime factorisation of 45470971.

Prime factorization of 45470971 = 13 × 13 × 7 × 7 × 17 × 17 × 19.

Factor Tree. A factor tree is a diagram used to determine the prime factors of a natural number greater than one.

The number 30 can be written as 6 × 5. Again, 6 can then be written as 2 × 3. Definitely, the number 30 can be factored different ways, but the result is always has one 2, one 3 and one 5. Below we will show a few factor tree for the number 30.

Example. Find the factor tree of 48.

Example. Find the factor tree of 60.

Exercise 1. Which of the following statements are true? Justify.

• (a) If two numbers are co-primes, at least one of them must be prime.
• (b) If a number is divisible by 3, it must be divisible by 9.
• (c) All numbers which are divisible by 4 must also be divisible by 8.
• (d) If a number exactly divides two numbers separately, it must exactly divide their sum.
• (e) If a number is divisible by 9, it must be divisible by 3.
• (f) All numbers which are divisible by 8 must also be divisible by 4.
• (g) If a number exactly divides the sum of two numbers, it must exactly divide the two numbers separately.
• (h) A number is divisible by 18, if it is divisible by both 3 and 6.
• (i) If a number is divisible by 9 and 10 both, then it must be divisible by 90.

Exercise 2. Which factors are not included in the prime factorisation of a composite number? Justify.

Exercise 3. Write the greatest four-digit number and express it in terms of its prime factors.

Exercise 4. Write the smallest five-digit number and express it in the form of its prime factors.

Exercise 5.  Find all the prime factors of 1729 and arrange them in ascending order. Now state the relation, if any; between two consecutive prime factors.

Exercise 6. The product of three consecutive numbers is always divisible by 6. Verify this statement with the help of some examples.

Exercise 7. The sum of two consecutive odd numbers is divisible by 4. Verify this statement with the help of some examples.

Exercise 8. In which of the following expressions, prime factorisation has been done?

• (a) 24 = 2 × 3 × 4
• (b) 54 = 2 × 3 × 9
• (c) 56 = 7 × 2 × 2 × 2
• (d) 70 = 2 × 5 × 7

Exercise 9. I am the smallest number, having four different prime factors. Can you find me?