Question

If the number \( 517*324 \) is completely divisible by 3, then the smallest whole number in the place of \(*\) will be ?

• A. \(2 \)
• B. \(3 \)
• C. \(4 \)
• D. \(5 \)

Hint:

Divisibility rule of 3: Sum up all the digits of the number. If the final sum is divisible by 3, then the entire number is divisible by 3. You can also eliminate digits that are already multiples of 3 (like 3) to speed up calculations!

Answer 1

The Correct Option is A

Concept: A number is completely divisible by 3 if and only if the sum of its individual digits is a multiple of 3 (or is divisible by 3). Let the missing digit in place of \(*\) be represented by the variable \(x\), where \(x \in \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}\).

Step 1: Write down the sum of the digits of the given number.
The given number is \( 517x324 \). Let us calculate the sum of all its digits in terms of \(x\): \[ \text{Sum of digits} = 5 + 1 + 7 + x + 3 + 2 + 4 \] Adding the constant numerical values together step-by-step: \[ 5 + 1 = 6 \] \[ 6 + 7 = 13 \] \[ 13 + 3 = 16 \] \[ 16 + 2 = 18 \] \[ 18 + 4 = 22 \] Thus, the total expression representing the sum of the digits is: \[ \text{Sum of digits} = 22 + x \]

Step 2: Find the smallest whole number value for \(x\).
For the number to be completely divisible by 3, the expression \((22 + x)\) must be an exact multiple of 3. Let us look at the multiples of 3 that are greater than or equal to 22: \[ 24, 27, 30, 33, \ldots \] To find the smallest whole number value for \(x\), we set the sum equal to the smallest possible multiple of 3 that is greater than or equal to 22, which is 24: \[ 22 + x = 24 \] Subtracting 22 from both sides of the equation yields: \[ x = 24 - 22 = 2 \] Since 2 is a single-digit whole number, it perfectly satisfies our condition. Hence, the smallest whole number in place of \(*\) is 2.