1. All about Factors K. Ganesan Grade Level: 5-8

2. Introduction What is a factor? F is factor of N if N / F has remainder 0. Example: The factors of 18 are 1, 2, 3, 6, 9, 18. So, 18 has 6 factors. How many factors does 360 have? Hmm, that is hard! It will take long time, Right? Can you think of a faster way to do this?

4. 360

5. Prime Factorization Prime numbers are those with only two factors: 1 and itself. Examples: 2 3 5 7 11 17 Any number can be written as a product of prime numbers. Given a number N, we can write N as p1^a * p2^b * p3^c etc Where p1, p2, p3 are some prime numbers. 360= 2^3 * 3^2 * 5 42 = 2 * 21 = 2 * 3 * 7

6. Counting of Factors Step 1: Determine the prime factors Step 2: Determine the power values of these prime numbers. Call them a, b, c etc The # of factors is then (a + 1) * (b + 1) * (c + 1) … Why? Example: 2^6 * 3^4 * 7^9 * 11^3 Any factor will have prime factors from this list. There are 7 choices for 2, 5 choices for 3, 10 choices for 7, and 4 choices for 11 So, 7 * 5 * 10 * 4 = 28 * 50 = 1400.

8. Sum of all factors What is the sum of all factors of 360? Do you need to find the factors and add them up? That would be painful and will take a long time? Is there a simple technique that will help us to get the sum quickly?

9. 360= 2^3 * 3^2 * 5