Finding Possible Values for P when 5306P2 is Divisible by 3

The problem asks us to find the difference between the squares of the greatest and smallest possible values of the digit P, such that the number 5306P2 is divisible by 3.

To solve this, we need to use the divisibility rule for the number 3.

Divisibility Rule for 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

Applying the Divisibility Rule to 5306P2

The given number is 5306P2. The digits are 5, 3, 0, 6, P, and 2.

Let's find the sum of these digits:

Sum of digits = $5 + 3 + 0 + 6 + P + 2$

Sum of digits = $16 + P$

For the number 5306P2 to be divisible by 3, the sum of its digits ($16 + P$) must be divisible by 3.

Finding Possible Values for Digit P

P is a single digit, which means P can be any integer from 0 to 9 (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).

We need to find which values of P make $16 + P$ divisible by 3. Let's test each possible value of P:

The possible values for P are the digits that make the sum $16 + P$ divisible by 3. From the table, these values are 2, 5, and 8.

• Possible values of P: 2, 5, 8

Identifying Greatest and Smallest Values of P

From the possible values (2, 5, 8):

• Greatest value of P is 8.
• Smallest value of P is 2.

Calculating the Difference Between Squares

We need to find the difference between the squares of the greatest value of P and the smallest value of P.

Greatest value squared = $8^2 = 8 \times 8 = 64$

Smallest value squared = $2^2 = 2 \times 2 = 4$

Difference between squares = (Greatest value)² - (Smallest value)²

Difference = $64 - 4$

Difference = $60$

Thus, the difference between the squares of the greatest and smallest values of P is 60.

Revision Table: Key Concepts Reviewed

Additional Information: Understanding Divisibility Rules

Divisibility rules are shortcuts to check if a number is divisible by another number without performing long division. The divisibility rule for 3 is based on the property that a number and the sum of its digits have the same remainder when divided by 3. This means if the sum of digits is divisible by 3 (remainder 0), the original number is also divisible by 3.

Let's consider why this rule works. Any number can be written in terms of powers of 10. For example, a three-digit number abc can be written as $100a + 10b + c$.

We can rewrite powers of 10 as follows:

• $1 = 0 \times 3 + 1$
• $10 = 3 \times 3 + 1$
• $100 = 33 \times 3 + 1$
• $1000 = 333 \times 3 + 1$

In general, $10^n = (\text{a multiple of } 3) + 1$. So, $10^n \equiv 1 \pmod{3}$.

Now, let's look at the number $5306P2$. This can be written as:

$5 \times 10^5 + 3 \times 10^4 + 0 \times 10^3 + 6 \times 10^2 + P \times 10^1 + 2 \times 10^0$

Modulo 3, this is equivalent to:

$5 \times 1 + 3 \times 1 + 0 \times 1 + 6 \times 1 + P \times 1 + 2 \times 1 \pmod{3}$

This simplifies to:

$5 + 3 + 0 + 6 + P + 2 \pmod{3}$

Which is the sum of the digits: $16 + P \pmod{3}$.

For the original number to be divisible by 3, its value modulo 3 must be 0. Therefore, the sum of its digits modulo 3 must also be 0, meaning the sum of digits is divisible by 3.

Other common divisibility rules:

• Divisibility by 2: The last digit is even (0, 2, 4, 6, 8).
• Divisibility by 4: The number formed by the last two digits is divisible by 4.
• Divisibility by 5: The last digit is 0 or 5.
• Divisibility by 6: The number is divisible by both 2 and 3.
• Divisibility by 9: The sum of the digits is divisible by 9.
• Divisibility by 10: The last digit is 0.