The product of the digits of a two-digit number is 36. If 45 is subtracted…

Concept: a two-digit number is not just a pair of symbols — it has place value. If its tens digit is a and its units digit is b, the number equals 10a + b, and the number formed by interchanging the two digits equals 10b + a. Subtracting the interchanged number from the original always leaves (10a + b) − (10b + a) = 9(a − b), so every digit-interchange difference is a multiple of 9 and fixes the difference of the digits.

Application: let the required number be 10a + b, with a as its tens digit and b as its units digit.

1. The product of the digits is 36, so a × b = 36.

2. Subtracting 45 interchanges the digits, so (10a + b) − 45 = 10b + a.

3. Rearranging that equation gives 9a − 9b = 45, hence a − b = 5.

4. The digit pairs whose product is 36 are 4 and 9, 6 and 6, and 9 and 4; of these only a = 9 with b = 4 gives a difference of 5.

5. Therefore the number is 10 × 9 + 4 = 94.

Cross-check: the digits of 94 multiply to 9 × 4 = 36, and 94 − 45 = 49, which is precisely 94 with its digits interchanged. Both conditions of the question are satisfied by 94.

Contrast with the values offered:

• 84: its digits multiply to 8 × 4 = 32, so the product condition is not met.

• 73: its digits multiply to 7 × 3 = 21, so the product condition is not met.

• 49: its digits do multiply to 4 × 9 = 36, but 49 − 45 = 4, while interchanging the digits of 49 would give 94.

The number that satisfies both conditions is 94, and 94 is not among the values offered, so the answer is None of the above.