18) Consider all four-digit numbers formed by the digits $3,4,6,7$, arranged in any order and without any digit being repeated. How many of these are divisible by 44?
(A) 0,
(B) 1,
(C) 2, (D) 3
(E) 4.
18. The answer is $(\mathbf{C})$.

A number is divisible by 44 if and only if it is divisible by both 4 and 11. In particular, the units digit must be even. A number is divisible by 11 if the difference between the sum of the digits in even positions and the sum of the digits in odd positions is a multiple of 11. In our case, the only possibility is that the difference between the two sums is zero, and since the units digit must be even, this happens only if the even digits are those of the units and hundreds places.

Only four situations remain possible: 3476, 3674, 7436, 7634, but we can still use the divisibility by four. A number is divisible by four if and only if the number formed by the units and tens digits is divisible by four. This condition is only satisfied by 3476 and 7436.