Find the number of all ordered pair (a, b) such that a ∈ A, b ∈ B, a is not equal to b and a + b is even [Read Full Q. below]

Question

Let A denote the set of all numbers between 1 and 700 which are divisible by 3 and let B denote the set of all numbers between 1 and 300 which are divisible by 7. Find the number of all ordered pair (a, b) such that a ∈ A, b ∈ B, a is not equal to b and a + b is even

My Approach

Okay so looking at the question, I can tell that I need to use the inclusion-exclusion principle but before applying it, I must dissect the sets. So a + b needs to be even this means that both of them (a,b) must be either even or odd. We can create a subset $p$ from $A$ which are all the even multiples of 3 till 300 and a subset $q$ for all odd multiples of 3. Then do the same for the set $B$ for the multiples of 7 and create subsets $m$, $n$ respectively.

Now we have 4 sets:

• $p$ = {even multiples of 3 till 300}
• $q$ = {odd multiples of 3 till 300}
• $m$ = {even multiples of 7 till 300}
• $n$ = {odd multiples of 7 till 300}

After which I apply the inclusion-exclusion principle to these sets $p ∪ q ∪ m ∪ n$ for which I got $84 - 252 + 189 - 84$ but that makes no sense since the answer would be in negative. There maybe a calculation error but anyway, if you have an easier approach please answer

Answer 1

I’ll get you most of the way and leave it to you to finish.

The largest and smallest multiples of $3$ in $A$ are $699=3\cdot233$ and $3=3\cdot1$, respectively, so there are $233$ multiples of $3$ in $A$; $116$ of them are even, and $117$ of them are odd.

The largest and smallest multiples of $7$ in $B$ are $294=7\cdot42$ and $7=7\cdot1$, respectively, so there are $42$ multiples of $7$ in $B$; $21$ of them are even, and $21$ of them are odd.

If $\langle a,b\rangle\in A\times B$, $a+b$ is even if and only if either $a$ and $b$ are both even, or $a$ and $b$ are both odd. There are $116\cdot21$ pairs with $a$ and $b$ both even and $117\cdot21$ with $a$ and $b$ both odd, and from this you can easily find the number of pairs such that $a+b$ is even.

However, that includes some pairs for which $a=b$, e.g., the pair $\langle 42,42\rangle$. How many such pairs are there?