What is the smallest whole number that should be added to 4456 so that the resulting sum is completely divisible by 6?

Introduction / Context:
This problem involves divisibility by 6 and requires us to adjust a given number slightly so that it satisfies the divisibility rules. A number is divisible by 6 if and only if it is divisible by both 2 and 3. The aim is to find the smallest non negative integer that, when added to 4456, makes the new total divisible by 6. This tests knowledge of basic divisibility rules and modular arithmetic.

Given Data / Assumptions:

- Original number is 4456. - We must add a smallest whole number k (k ≥ 0) such that 4456 + k is divisible by 6. - A number is divisible by 6 if it is divisible by both 2 and 3.

Concept / Approach:
First check whether 4456 is divisible by 2 and by 3. Divisibility by 2 depends on the last digit being even, while divisibility by 3 depends on the sum of digits being a multiple of 3. If the number already meets one of the conditions, we only need to adjust it to satisfy the other while keeping the first condition valid. We search for the smallest k such that the new number is divisible by both 2 and 3.

Step-by-Step Solution:
Step 1: Check divisibility of 4456 by 2. The last digit is 6, which is even, so 4456 is divisible by 2. Step 2: Check divisibility of 4456 by 3. Sum of digits is 4 + 4 + 5 + 6 = 19. Step 3: Since 19 is not a multiple of 3, 4456 is not divisible by 3. Step 4: Let k be the required smallest non negative integer such that 4456 + k is divisible by 6. Step 5: For divisibility by 3, we need (sum of digits of 4456) + (sum of digits of k) to be a multiple of 3. In modular arithmetic, 19 + k ≡ 0 (mod 3). Step 6: Since 19 ≡ 1 (mod 3), we need k ≡ 2 (mod 3). Step 7: For divisibility by 2, 4456 + k must be even. Because 4456 is even, k must also be even so that the sum remains even. Step 8: The smallest non negative integer that is both even and congruent to 2 modulo 3 is k = 2. Step 9: Check: 4456 + 2 = 4458. Sum of digits is 4 + 4 + 5 + 8 = 21, which is divisible by 3, and the last digit 8 is even. So 4458 is divisible by 6.

Verification / Alternative check:
We can also test options directly. Adding 1 gives 4457, which is odd and not divisible by 2. Adding 2 gives 4458, which we saw is divisible by both 2 and 3. Adding 3 gives 4459, which is odd. Adding 4 gives 4460, which is even, but the sum of digits 4 + 4 + 6 + 0 = 14 is not a multiple of 3. Therefore 2 is indeed the smallest suitable number.

Why Other Options Are Wrong:
Option D (1) and option B (3) produce odd results, failing the divisibility by 2 condition. Option A (4) produces a number that is even but not divisible by 3. Therefore they do not give a sum divisible by 6.

Common Pitfalls:
Some learners may only check divisibility by 3, forgetting that divisibility by 6 requires both 2 and 3. Others may choose the smallest k that fixes one condition but breaks the other. Always verify both conditions for divisibility by 6 to avoid such mistakes.

Final Answer:
The smallest number to be added is 2, which corresponds to option C.