Every digit of a natural number is either 3 or 4. The number is divisible by…

Concept

A number is divisible by 4 when the number formed by its last two digits is divisible by 4.

A number is divisible by 3 when the sum of its digits is divisible by 3. A number satisfying both tests is divisible by 12.

Application

1. A one-digit candidate can only be 3 or 4; neither is divisible by both 3 and 4.

2. Among the two-digit numbers made from 3 and 4, only 44 has a last-two-digit value divisible by 4. Its digit sum is 8, which is not divisible by 3.

3. A three-digit candidate passing the divisibility-by-4 test must end in 44, so the candidates in increasing order are 344 and 444. Their digit sums are 11 and 12 respectively.

4. The sum 12 is divisible by 3, whereas 11 is not. Therefore 444 is the first candidate that passes both divisibility tests.

Cross-check

Direct division gives 444 / 12 = 37, so 444 is divisible by both 3 and 4. The shorter candidate 44 fails the divisibility-by-3 test, confirming minimality.

Hence, the smallest such number is 444.