Algebraic Expressions Divisible by 3

Problem

If a+2 is divisible by 3, which of the following expressions is/are also divisible by 3?

A. 3+2a

B. 5a-2

C. 8+7a

D. 1+5a

Solution

When a number is divisible by 3 then it is a multiple of 3. So, if it is given that a+2 is divisible by 3 then a+2 is a multiple of 3. Hence, we can write a+2 = 3N, where N is an integer. This further implies that a=3N-2. So, to test if an expression in a is divisible by 3, we need to substitute 3N-2 to a and see if the resulting expression is a multiple of 3.

Is 3+ 2a divisible by 3?

Answer: Nope. This is because 3+2a=3+2(3N-2)= 6N-1. 6N-1 is not a multiple of 3.

Is 5a-2 divisible by 3? Let’s see. 5a-2=5(3N-2)-2=15N-12. Is 15N-12 divisible by 3? Definitely. 15N-12=3(5N-3).

I will leave it to you to test if 8+7a and 1+5a are also divisible by 3.

Challenge

1. Which among the choices, A, B, C, and D are also divisible by 2?
2. Which among the choices, A, B, C, and D are also divisible by 5?
3. Construct an expression in a  that is both divisible by 2 and 5.
4. Make a problem similar to the given problem that involves divisibility and algebraic expressions.