BTGmoderatorDC wrote:Let abcd be a general four-digit number. How many odd four-digits numbers abcd exist such that the four digits are all distinct, no digit is zero, and the product of a and b is the two-digit number cd?

(A) 4
(B) 6
(C) 12
(D) 24
(E) 36

Condition:
a*b = integer cd

Since abcd must be odd and composed of 4 distinct digits, we get the following options for integer cd:
21, 31, 41, 51, 61, 71, 81, 91
13, 23, 43, 53, 63, 73, 83, 93
15, 25, 35, 45, 65, 75, 85, 95
17, 27, 37, 47, 57, 67, 87, 97
19, 29, 39, 49, 59, 69, 79, 89

Only the options in blue can be equal to the product of two distinct digits a and b:
Case 1: cd = 21, with the result that a=3 and b=7 or a=7 and b=3
In this case, abcd = 3721 or 7321
Case 2: cd = 63, with the result that a=7 and b=9 or a=9 and b=7
In this case, abcd = 7963 or 9763
Case 3: cd = 27, with the result that a=3 and b=9 or a=9 and b=3
In this case, abcd = 3927 or 9327

Total options for abcd = 6.

The correct answer is B.