Find out the smallest number which should replace x in the given number 296x12…

A number is divisible by 12 exactly when it is divisible by both 3 and 4 (since 12 = 3 x 4, and 3 and 4 share no common factor). The rule for divisibility by 4 checks only the last two digits of the number, while the rule for divisibility by 3 checks whether the sum of all digits is a multiple of 3.

Applying this to 296x12:

1. The last two digits of the number are fixed at 12, and 12 divided by 4 = 3 exactly - so the number is divisible by 4 for every value of x.

2. So only the divisibility-by-3 condition depends on x. The digit sum is 2 + 9 + 6 + x + 1 + 2 = 20 + x.

3. For the number to be divisible by 3, (20 + x) must be a multiple of 3. Testing digits from the smallest (x = 0) upward: x = 0 gives 20 (not a multiple of 3); x = 1 gives 21 = 3 x 7 (a multiple of 3).

4. So x = 1 is the smallest digit for which 20 + x is divisible by 3.

Check: with x = 1 the number is 296112. 296112 divided by 12 = 24676 exactly (24676 x 12 = 296112), confirming it is divisible by 12.

Hence, the smallest digit that replaces x is 1.