Divisibility Analysis of Rearranged Digits

The question asks us to determine a number that will always divide a new number formed by rearranging the digits of 468312 in descending order. We are given that the original number, 468312, is divisible by 57.

Checking Initial Information

First, let's verify the given information. The number is 468312. The divisor is 57. Note that $57 = 3 \times 19$. A number is divisible by 57 if it is divisible by both 3 and 19.

• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3. Sum of digits of 468312 = $4 + 6 + 8 + 3 + 1 + 2 = 24$. Since 24 is divisible by 3 ($24 \div 3 = 8$), the number 468312 is divisible by 3.
• Divisibility by 19: We can check this by division: $468312 \div 19 = 24648$. So, 468312 is divisible by 19.

Since 468312 is divisible by both 3 and 19, it is indeed divisible by 57. ($468312 \div 57 = 8216$).

Rearranging the Digits

The digits in the number 468312 are 4, 6, 8, 3, 1, 2. Rearranging these digits in descending order gives the number 864321.

Testing Divisibility of the Rearranged Number

We need to find which of the given options (2, 3, 19, 17) will always divide the rearranged number 864321. The key is that the divisibility property must hold true for *any* rearrangement, but specifically, we are checking the descending order rearrangement.

• Option 1: Divisibility by 2 The rearranged number is 864321. It ends in the digit 1 (an odd number). Therefore, 864321 is not divisible by 2.
• Option 2: Divisibility by 3 The sum of the digits of the rearranged number 864321 is $8 + 6 + 4 + 3 + 2 + 1 = 24$. As established earlier, any number formed by rearranging the digits 4, 6, 8, 3, 1, 2 will have a digit sum of 24. Since 24 is divisible by 3, any such rearranged number will always be divisible by 3. Let's check our specific rearrangement: $864321 \div 3 = 288107$. Thus, 864321 is divisible by 3.
• Option 3: Divisibility by 19 Let's check if the rearranged number 864321 is divisible by 19. $864321 \div 19 \approx 45490.58$. It is not exactly divisible by 19.
• Option 4: Divisibility by 17 Let's check if the rearranged number 864321 is divisible by 17. $864321 \div 17 \approx 50842.41$. It is not exactly divisible by 17.

Conclusion on Divisibility

Based on the analysis, the only number from the options that is guaranteed to divide the number formed by rearranging the digits of 468312 in descending order (864321) is 3. This is because the divisibility by 3 depends solely on the sum of the digits, which remains constant regardless of the order of the digits.