Final answer:
The two-digit number sought for is 64, which has a product of digits equal to 24, and when 18 is subtracted from it, the digits interchange their places.
Step-by-step explanation:
The question requires finding a two-digit number where the product of its digits is 24 and, when 18 is subtracted from the number, the digits interchange their places. First, we list possible pairs of digits that yield a product of 24: (1, 24), (2, 12), (3, 8), and (4, 6). Since we need a two-digit number, the valid pairs are (3, 8) and (4, 6). Now, let's investigate which pair fulfills the condition of digit interchange upon subtracting 18.
Assuming that the original number is 10x + y (where x is the tens digit and y is the units digit), and after the subtraction of 18, the number becomes 10y + x. We set up the equation:
10x + y - 18 = 10y + x
By simplifying, we get:
9x - 9y = 18
This simplifies to:
x - y = 2
Now we substitute the pairs (3, 8) and (4, 6) to see which one satisfies the equation. The pair (3, 8) does not satisfy it, but (4, 6) does because:
4 - 6 = -2
However, we want the positive difference, so if the original number is the reversed digits, 64, subtracting 18 will give us 46. Therefore, the two digit number we are seeking is 64.