If the number 23P62971335 is divisible by the smallest odd composite number, then what is the value of P ?
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Understanding the Problem: Divisibility by Smallest Odd Composite Number
The question asks us to find the value of the digit P in the number 23P62971335, given that this number is divisible by the smallest odd composite number. To solve this, we first need to identify what the smallest odd composite number is and then use the divisibility rule for that number.
Identifying the Smallest Odd Composite Number
Let's break down the term "smallest odd composite number":
- Composite Number: A natural number that has more than two distinct positive divisors (factors). In other words, it's a number that is not prime (excluding 1). Examples: 4, 6, 8, 9, 10, 12, ...
- Odd Number: An integer that is not divisible by 2. Examples: 1, 3, 5, 7, 9, 11, ...
Now let's list the first few natural numbers and categorize them:
| Number | Type (Odd/Even) | Type (Prime/Composite/Neither) | Notes |
|---|---|---|---|
| 1 | Odd | Neither | Only one factor (1) |
| 2 | Even | Prime | Factors: 1, 2 |
| 3 | Odd | Prime | Factors: 1, 3 |
| 4 | Even | Composite | Factors: 1, 2, 4 |
| 5 | Odd | Prime | Factors: 1, 5 |
| 6 | Even | Composite | Factors: 1, 2, 3, 6 |
| 7 | Odd | Prime | Factors: 1, 7 |
| 8 | Even | Composite | Factors: 1, 2, 4, 8 |
| 9 | Odd | Composite | Factors: 1, 3, 9 |
Looking at the list, the odd numbers are 1, 3, 5, 7, 9, ... Among these, we need the smallest one that is also composite. From our table, we see that 9 is the first number that is both odd and composite.
Therefore, the smallest odd composite number is 9.
Applying the Divisibility Rule for 9
The problem states the number 23P62971335 is divisible by the smallest odd composite number, which is 9. The divisibility rule for 9 states that a number is divisible by 9 if and only if the sum of its digits is divisible by 9.
Calculating the Sum of the Digits
The number is 23P62971335. The digits are 2, 3, P, 6, 2, 9, 7, 1, 3, 3, and 5.
Let's sum the known digits:
Sum of known digits \(= 2 + 3 + 6 + 2 + 9 + 7 + 1 + 3 + 3 + 5\)
Sum of known digits \(= 5 + 6 + 2 + 9 + 7 + 1 + 3 + 3 + 5\)
Sum of known digits \(= 11 + 2 + 9 + 7 + 1 + 3 + 3 + 5\)
Sum of known digits \(= 13 + 9 + 7 + 1 + 3 + 3 + 5\)
Sum of known digits \(= 22 + 7 + 1 + 3 + 3 + 5\)
Sum of known digits \(= 29 + 1 + 3 + 3 + 5\)
Sum of known digits \(= 30 + 3 + 3 + 5\)
Sum of known digits \(= 33 + 3 + 5\)
Sum of known digits \(= 36 + 5\)
Sum of known digits \(= 41\)
The total sum of the digits of the number 23P62971335 is \(41 + P\).
Finding the Value of P
For the number to be divisible by 9, the sum of its digits (\(41 + P\)) must be a multiple of 9. The digit P can be any integer from 0 to 9 (since it's a single digit in the number).
Let's check the possible values for \(41 + P\) that are multiples of 9:
- The multiples of 9 are ..., 36, 45, 54, 63, ...
- We need to find a multiple of 9 that is close to 41.
If \(41 + P = 36\), then \(P = 36 - 41 = -5\). This is not a valid digit.
If \(41 + P = 45\), then \(P = 45 - 41 = 4\). This is a valid digit (0-9).
If \(41 + P = 54\), then \(P = 54 - 41 = 13\). This is not a valid single digit.
Since P must be a single digit between 0 and 9, the only possible value for P is 4.
Verification
If P = 4, the number becomes 23462971335. The sum of digits is \(41 + 4 = 45\). Since 45 is divisible by 9 (\(45 \div 9 = 5\)), the number 23462971335 is indeed divisible by 9, which is the smallest odd composite number.
Thus, the value of P is 4.
Revision Table: Key Concepts
| Concept | Definition/Rule | Relevance to Problem |
|---|---|---|
| Smallest Odd Composite Number | The smallest natural number that is both odd and has more than two factors. It is 9. | The divisor given in the problem. |
| Divisibility Rule for 9 | A number is divisible by 9 if the sum of its digits is divisible by 9. | Method used to find the missing digit P. |
| Sum of Digits | Adding all the digits of a number together. | Crucial calculation for applying the divisibility rule of 9. |
Additional Information: Number Properties
Understanding different types of numbers and their properties is fundamental in number theory. Here are some related concepts:
- Prime Numbers: Natural numbers greater than 1 that have exactly two distinct positive divisors: 1 and themselves. Examples: 2, 3, 5, 7, 11, 13, ...
- Even Numbers: Integers that are divisible by 2. They can be written in the form \(2k\), where \(k\) is an integer. Examples: ..., -4, -2, 0, 2, 4, ...
- Divisibility Rules: Shortcuts to determine if a number is divisible by another number without performing long division. For example, a number is divisible by 3 if the sum of its digits is divisible by 3; a number is divisible by 5 if its last digit is 0 or 5.
This problem combined the identification of a specific type of number (smallest odd composite) with the application of a common divisibility rule (for 9).
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select the correct answer using the code given below:
