If A,BCD is a four-digit positive integer such that A,BCD is equal to the product of the two-digit number AB and the three digit number ABC, which of the following two-digit numbers cannot be a factor of A,BCD? (Note: the first digit of a number cannot be equal to 0.)
I. AD
II. CD
III. CA Choose one:
A) None
B) II only
C) III only
D) II and III only
E) I, II, and III
Ideally, ABC * AB = ABCD should make you think of 1s and 0s. The digits needn't be unique so the first thing that comes to mind is   100*10= 
<gadget id="calculator">  100*10</gadget>
<output>1_000</output>
1000. But since CA and CD are given as two digit numbers, next you should think of   101*10= 
<gadget id="calculator">  101*10</gadget>
<output>1_010</output>
1010 or any such number. This satisfies all conditions so we can analyse using this example.   101 * 10= 
<gadget id="calculator">101 * 10</gadget>
<output>1_010</output>
1010   AD= 10 - a factor of 1010   CD = 10 - a factor of 1010   CA = 11 - not a factor of 1010   Now the only thing left to figure is whether it is possible that C1 is a factor of 10C0 for some value of C.   10C0 = 1000 + C0 = 999 + C1   C1 is divisible by C1 but we don't know whether 999 is divisible by C1.   999 = 3^3*37   It has no two digit factor ending in 1 so it will not be divisible by C1.   Answer (III only)
<result>C</result>
If A,BCD is a four-digit positive integer such that A,BCD is equal to the product of the two-digit number AB and the three digit number ABC, which of the following two-digit numbers cannot be a factor of A,BCD? (Note: the first digit of a number cannot be equal to 0.)
I. AD
II. CD
III. CA