Let p be a two-digit number and q be the number consisting of same digits written in reverse order. If p × q = 2430, then what is the difference between p and q?

The correct option is 9.

Let the two-digit number $p$ be represented as:

$p=10a+b$

where $a$ and $b$ are digits such that $1\le a\le 9$ and $0\le b\le 9$.

The number $q$ formed by reversing the digits of $p$ is:

$q=10b+a$

We are given that:

$p\times q=2430$

First, let us find the prime factorization of 2430 to determine the possible values for $p$ and $q$:

$2430=2\times 3^5\times 5=2\times 243\times 5$

Since $p$ and $q$ are both two-digit numbers, their product ends in 0, which means one of the numbers must end in 5 or 0. If a two-digit number ends in 0, its reverse would start with 0 (making it a single-digit number), which is not the case here since both are two-digit numbers. Therefore, one of the numbers must end in 5.

Let $p$ be a multiple of 5 ending in 5. Thus, $p$ can be written as $10a+5$.

Then $q$ (its reverse) must start with 5, so $q=50+a$.

Testing factors of 2430 around 50:

$2430÷54=45$

Notice that:

$p=45$

$q=54$

Here, $q$ is indeed the reverse of $p$, and their product is:

$45\times 54=2430$

Now, we find the absolute difference between $p$ and $q$:

$|p-q|=|54-45|=9$

Thus, the difference between $p$ and $q$ is 9.