P and Q are two two-digit numbers Their product equals

P = xy ⇒ 10x + y
Q = pq ⇒ 10p + q

The reverse numbers are

Reverse(P) = yx = 10y + x
Reverse(Q) = qp = 10q + p

This question can be solved using a two-step process -

1. Identify the relationship between x, y, p, and q.
2. Identify the pairs of values that satisfy that relationship

Step 1: Identify the relationship between x, y, p, and q.



\((10x + y) (10p+q) = (10q + p)(10y +x)\)

\(100xp + 10xq + 10yq + yq= 100qy + 10qx + 10py + px\)

\(99(xp - yq) = 0\)

\(x*p =y*q \)

Step 2: Identify the pairs of values that satisfy that relationship



Inferences:

• None of the digits in P or Q is equal to the other digit in it or any digit in the other number
  \(x \neq y \neq p \neq q\)
• The product of tens digits of the two numbers' is a composite single digit number
  \(x *p\) is composite number
  If x = 1, p \(\neq\) prime and vice-versa
  

Let's try to understand the given conditions a bit better, let's say x = 2 and p = 3

2 * 3 = y * q

So (y,q) can be (1,6) or (6,1) but not (2,3) or (3,2). So for one combination of (x,p) we can have four possible values

(x,p) = (2,3)

• (y,p) = (6,1)
  xy = 26 | pq = 31
  
• (y,p) = (1,6)
  xy = 21 | pq = 36

(x,p) = (3,2)

• (y,p) = (6,1)
  xy = 36 | pq = 21
  
• (y,p) = (1,6)
  xy = 31 | pq = 26

Possible values of (x,p) = (1,6), (1,8), (2,3), and (4,2).

Total possible combinations = 4 * 4 =16

Option B

P.S. - This is not a GMAT Style question.
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Let's say the 2 digit n.o are in the for of ab cd

So value of the numbers is 10a+b, 10c+d.

If we solve the below equation i.e, (10a+b)(10c+d) = (10b+a)(10d+c)
We will get ac = bd.

We know that ac is a composite single digit. So possible values are 4,6,8,9


a*c = b*d = {4,6,8,9}

All 4 are distinct.

So 4,9 are not possible.

a*c=6 or 8 .

2*2*2*2 possibilities.

16
IMO B

What is the source of the qs? Is this OG?