Understanding Divisibility by 15

For an 8-digit number like 136p5785 to be divisible by 15, it must satisfy the divisibility rules for its prime factors. The prime factors of 15 are 3 and 5. Therefore, the number must be divisible by both 3 and 5.

Divisibility Rule for 5

A number is divisible by 5 if its last digit is either 0 or 5.

The given number is 136p5785. The last digit of this number is 5. Since the last digit is 5, the number 136p5785 is already divisible by 5, regardless of the value of the digit 'p'. This condition is satisfied.

Divisibility Rule for 3

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

The digits in the number 136p5785 are 1, 3, 6, p, 5, 7, 8, and 5. Let's find the sum of these digits:

Sum of digits = \(1 + 3 + 6 + p + 5 + 7 + 8 + 5\)

Let's sum the known digits: \(1 + 3 + 6 + 5 + 7 + 8 + 5 = 35\)

So, the total sum of the digits is \(35 + p\).

For the number 136p5785 to be divisible by 3, the sum of its digits, \(35 + p\), must be divisible by 3.

Finding Possible Values for Digit p

The digit 'p' is in the thousands place of the 8-digit number. A digit can be any whole number from 0 to 9. We need to find which values of 'p' (from 0 to 9) make \(35 + p\) divisible by 3.

We can test each possible value of 'p':

• If \(p = 0\), sum = \(35 + 0 = 35\). \(35 \div 3\) is not an integer.
• If \(p = 1\), sum = \(35 + 1 = 36\). \(36 \div 3 = 12\). This is divisible by 3.
• If \(p = 2\), sum = \(35 + 2 = 37\). \(37 \div 3\) is not an integer.
• If \(p = 3\), sum = \(35 + 3 = 38\). \(38 \div 3\) is not an integer.
• If \(p = 4\), sum = \(35 + 4 = 39\). \(39 \div 3 = 13\). This is divisible by 3.
• If \(p = 5\), sum = \(35 + 5 = 40\). \(40 \div 3\) is not an integer.
• If \(p = 6\), sum = \(35 + 6 = 41\). \(41 \div 3\) is not an integer.
• If \(p = 7\), sum = \(35 + 7 = 42\). \(42 \div 3 = 14\). This is divisible by 3.
• If \(p = 8\), sum = \(35 + 8 = 43\). \(43 \div 3\) is not an integer.
• If \(p = 9\), sum = \(35 + 9 = 44\). \(44 \div 3\) is not an integer.

The possible values for 'p' that make the number divisible by 3 are 1, 4, and 7.

Since the number is divisible by 5 for any digit 'p', the values of 'p' that make the number divisible by 15 are those that make it divisible by 3.

Thus, the possible values for 'p' are 1, 4, and 7.

Finding the Least Possible Value of P

The question asks for the least possible value of P. Comparing the possible values 1, 4, and 7, the smallest value is 1.

Therefore, the least possible value of P is 1.

Revision Table: Key Divisibility Rules

Additional Information: Divisibility Concepts

Divisibility rules are useful shortcuts to determine if one number can be divided by another without performing the full division calculation. Understanding these rules, especially for small prime numbers, helps in solving problems related to factors, multiples, and number properties. For composite numbers like 15, the divisibility rule is derived from the rules of its prime factors (3 and 5). A number is divisible by a composite number if and only if it is divisible by each of its relatively prime factors. Since 3 and 5 are prime and distinct, they are relatively prime.