Parities and the four common math operators

Number Theory Level 2

Let $a$ and $b$ be integers such that $a+b,\quad a-b,\quad a\times b,\quad a\div b$ are all integers as well.

If $a+b$ is an even number, then which of the following must be an even number as well?

Select one or more

a-b

a\times b

a\div b

2 solutions

David Vreken
Oct 23, 2018

If $a + b$ is an even number, then either (1) both $a$ and $b$ are odd, or (2) both $a$ and $b$ are even.

(1) If both $a$ and $b$ are odd (for example, $a = 15$ and $b = 3$ ), then $a - b$ is even, $a \times b$ is odd, and $a \div b$ is odd.

(2) If both $a$ and $b$ are even (for example, $a = 8$ and $b = 4$ ), then $a - b$ is even, $a \times b$ is even, and $a \div b$ is even.

In either case, $a - b$ is even, but $a \times b$ and $a \div b$ could either be odd or even.

@Pi Han Goh How do we add multiple option correct type questions?

Aaryan Maheshwari - 2 years, 7 months ago

The staffs did it, not me.

Pi Han Goh - 2 years, 7 months ago

Jon Haussmann
Oct 31, 2018

Since $a + b$ and $2b$ are even numbers, $(a + b) - 2b = a - b$ must be even as well.

The example $a = b = 1$ shows that $a \times b$ and $a \div b$ are not necessarily even.