Question

Let A be the set of all composites less than 10, and B be the set of positive even integers less than 10. How many different sums of the form a + b are possible if a is in A and b is in B?

Answer 1

16 different forms of `a+b`. 10 unique sums.

Explanation:

The set `bb(A)`

A composite is a number that can be divided evenly by a smaller number other than 1. For instance, 9 is composite `(9/3=3)` but 7 is not (another way of saying this is a composite number is not prime). This all means that the set `A` consists of:

`A={4,6,8,9}`

The set `bb(B)`

`B={2,4,6,8}`

We're now asked for the number of different sums in the form of `a+b` where `a in A, b in B`.

In one reading of this problem, I'd say there are 16 different forms of `a+b` (with things like `4+6` being different than `6+4`).

However, if read as "How many unique sums are there?", perhaps the easiest way to find that is to table it out. I'll label the `a` with `color(red)("red")` and `b` with `color(blue)("blue")`:

`(("",color(blue)2,color(blue)4,color(blue)6,color(blue)8),(color(red)4,6,8,10,12),(color(red)6,8,10,12,14),(color(red)8,10,12,14,16),(color(red)9,11,13,15,17))`

And so there are 10 unique sums: `6, 8, 10, 11, 12, 13, 14, 15, 16, 17`