Difficulty: Easy
Correct Answer: 2
Explanation:
Introduction:
This is a basic divisibility test problem with a missing digit. You are given a large number with one unknown digit and asked to choose that digit so the entire number becomes divisible by 3. Such problems train you to apply divisibility rules quickly without long division.
Given Data / Assumptions:
Concept / Approach:
The rule for divisibility by 3 states that a number is divisible by 3 if and only if the sum of its digits is divisible by 3. We therefore sum the known digits, include the unknown digit x in the sum, and choose x so that the total is a multiple of 3.
Step-by-Step Solution:
Step 1: Let the missing digit be x.Digits are 5, 1, 7, x, 3, 2, 4.Step 2: Sum the known digits.5 + 1 + 7 + 3 + 2 + 4 = 22.Step 3: Include x in the total sum.Total sum = 22 + x.Step 4: Apply the divisibility by 3 rule.We need 22 + x to be a multiple of 3.The next multiples of 3 after 22 are 24, 27, 30, etc.So 22 + x = 24, 27, 30, ...This gives x = 2, 5, 8, ...Step 5: Choose the smallest whole number.The smallest x in {2, 5, 8, ...} is 2.
Verification / Alternative check:
Substitute x = 2 to get the number 5172324. Sum of digits = 5 + 1 + 7 + 2 + 3 + 2 + 4 = 24, which is divisible by 3. Therefore, the full number is divisible by 3 as required.
Why Other Options Are Wrong:
0 and 1 make the digit sum 22 and 23 respectively, which are not multiples of 3. 5 and 8 also give valid divisibility by 3, but they are larger than 2. The question explicitly asks for the smallest such digit, so 2 is the correct choice.
Common Pitfalls:
Some learners forget the “smallest” condition and might choose any digit that works, such as 5 or 8. Others try full long division rather than using the far quicker digit-sum rule.
Final Answer:
The smallest whole number that can replace * is 2.
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