3 In class on Wednesday Jan 14th we talked about divisibilit

3. In class on Wednesday, Jan 14th, we talked about divisibility rules. a. Explain why the rule for 4 works: To determine if a number is divisible by 4, we check to see if the last two digits are a multiple of 4. (ex: 89623452 is divisible by 4 because 52 is a multiple of 4. b. Explain why the rule for 3 works: To determine if a number is divisible by 3, we check to see if the sum of the digits is a multiple of 3. (ex: 111111 is divisible by 3 because l+1+1+1+1+1 = 6 which is a multiple of 3

Solution

a. For any integer x written as a_na_n-1a_n-2...a₁a₀, we will show that x is divisible by 4 if a₁a₀ is divisible by 4.

If we write x as a_na_n-1a_n-2...a₁a₀, then we can also write:

x= a₀ + a₁(10) + a₂(10² )+ a₃(10³)... + a_n-2(10^n-2)+a_n-1(10^n-1)+ a_n(10ⁿ)

                 = (a_n