In the number 835x4, by replacing ‘x’ with which of the following digits will…

Key idea: A number divisible by 12 must be divisible by both 3 and 4.

1. Check divisibility by 4: Look at the last two digits, which are x4. The two-digit numbers divisible by 4 among 04, 14, 24, ..., 94 are 04, 24, 44, 64, 84, so x can be 0, 2, 4, 6, or 8.

2. Check divisibility by 3: Sum the digits: 8 + 3 + 5 + x + 4 = 20 + x. For divisibility by 3 we need 20 + x to be a multiple of 3, so x = 1, 4, or 7.

Intersection: The digits that satisfy both conditions are the common values of {0,2,4,6,8} and {1,4,7}, which is x = 4.

Final check: Replacing x with 4 gives 83544. 44 is divisible by 4 and the digit sum 24 is divisible by 3, so 83544 is divisible by 12 (83544 ÷ 12 = 6962).

Why the other choices fail:

• Digit 1: The digit sum 21 is divisible by 3, but the last two digits 14 are not divisible by 4, so the whole number is not divisible by 12.

• Digit 7: The digit sum 27 is divisible by 3, but the last two digits 74 are not divisible by 4.

• Digit 8: The last two digits 84 are divisible by 4, but the digit sum 28 is not divisible by 3.