Complete 54?? to Make It Divisible by 6: Missing Digit Challenge

Question

Complete the number so that it is divisible by 6 without a remainder:

54?? 54??

Video Solution

Solution Steps

00:00 Complete the missing digits so that the number is divisible by 6
00:04 For a number to be divisible by 6, it must be even
00:08 and the sum of its digits must be divisible by 3
00:13 Let's check each option and eliminate the unsuitable ones
01:03 And this is the solution to the question

Step-by-Step Solution

To solve this problem, we need to find digits to replace the question marks in 54?? 54?? such that the resulting number is divisible by 6. We'll use the divisibility rules for both 2 and 3.

First, since the number must be even, the last digit must be one of the even numbers: 0, 2, 4, 6, or 8.

Second, we need the sum of the digits to be divisible by 3. The initial digits '54' have a sum of 5+4=9 5 + 4 = 9 . Since 9 is already divisible by 3, we need the sum of the missing digits (let’s denote them as x and y) to also result in a number divisible by 3 when added to 9.

Therefore, we need 9+x+y0(mod3) 9 + x + y \equiv 0 \pmod{3} . This simplifies to x+y0(mod3) x + y \equiv 0 \pmod{3} .

Let us check with the provided options:

  • Option 1: 3,3 3,3 : The sum x+y=3+3=6 x + y = 3 + 3 = 6 . Divisible by 3. Last digit is 3 (odd), this does not satisfy divisibility by 2.
  • Option 2: 6,8 6,8 : The sum x+y=6+8=14 x + y = 6 + 8 = 14 . Not divisible by 3.
  • Option 3: 5,5 5,5 : The sum x+y=5+5=10 x + y = 5 + 5 = 10 . Not divisible by 3.
  • Option 4: 3,1 3,1 : The sum x+y=3+1=4 x + y = 3 + 1 = 4 . Not divisible by 3.

Since none of the provided options satisfy both divisibility conditions, the correct answer is "None of the above".

In conclusion, Option 5: "None of the above" is the correct choice.

Answer

None of the above