Finding the Values of p and q in a 7-Digit Number Divisible by 75

We are given a 7-digit number 678p37q and told that it is divisible by 75. We are also given the condition that 'p' is not a composite number. We need to find the values of the digits 'p' and 'q'.

A number is divisible by 75 if and only if it is divisible by both 3 and 25. This is because 75 can be factored into its prime factors as $3 \times 25 = 3 \times 5^2$. Since 3 and 25 are coprime (their greatest common divisor is 1), the divisibility by 75 depends on divisibility by both 3 and 25.

Step 1: Apply Divisibility Rule for 25

For a number to be divisible by 25, the number formed by its last two digits must be divisible by 25. In the given number 678p37q, the last two digits form the number 7q.

The possible two-digit numbers ending in any digit 'q' that are divisible by 25 are 00, 25, 50, and 75.

Looking at the last two digits, 7q, we need this number to be one of these multiples of 25. The only possibility where the tens digit is 7 is 75.

Therefore, the number 7q must be 75. This implies that the digit q must be 5.

So, the number becomes 678p375.

Step 2: Apply Divisibility Rule for 3

For a number to be divisible by 3, the sum of its digits must be divisible by 3. The number is 678p375. Let's find the sum of its digits:

Sum of digits = $6 + 7 + 8 + p + 3 + 7 + 5$

Sum of digits = $36 + p$

For the number to be divisible by 3, the sum of the digits, $36 + p$, must be divisible by 3. Since 36 is divisible by 3, $36+p$ will be divisible by 3 if and only if 'p' is divisible by 3.

The digit 'p' can be any single digit from 0 to 9. The single digits that are divisible by 3 are 0, 3, 6, and 9.

Step 3: Apply the Condition on 'p'

We are given that 'p' is not a composite number. Let's review what a composite number is.

• A composite number is a positive integer that has at least one divisor other than 1 and itself.
• Prime numbers are positive integers greater than 1 that have no positive divisors other than 1 and themselves (e.g., 2, 3, 5, 7, 11...).
• The number 1 is neither prime nor composite.
• The number 0 is also not considered composite.

The possible single digits for 'p' (0-9) are:

So, the single digits that are not composite are 0, 1, 2, 3, 5, 7.

Step 4: Combine Conditions to Find 'p' and 'q'

From Step 1, we know that $q = 5$.

From Step 2, we know that 'p' must be one of {0, 3, 6, 9}.

From Step 3, we know that 'p' must be one of {0, 1, 2, 3, 5, 7} (not composite).

We need the value of 'p' that satisfies both conditions. Looking for the intersection of the two sets {0, 3, 6, 9} and {0, 1, 2, 3, 5, 7}, the common values are 0 and 3.

So, possible values for (p, q) are (0, 5) and (3, 5).

Step 5: Check the Options

Let's look at the given options:

1. $p=5, q=5$: Here $p=5$. 5 is not divisible by 3. Sum of digits $36+5=41$, not divisible by 3. This option is incorrect.
2. $p=3, q=0$: Here $q=0$. Last two digits are 70, which is not divisible by 25. This option is incorrect.
3. $p=3, q=5$: Here $q=5$. Last two digits are 75, divisible by 25. Here $p=3$. 3 is divisible by 3 and is not a composite number. Sum of digits $36+3=39$, divisible by 3. This option satisfies all conditions.
4. $p=2, q=5$: Here $q=5$. Last two digits are 75, divisible by 25. Here $p=2$. 2 is not divisible by 3. Sum of digits $36+2=38$, not divisible by 3. This option is incorrect.

The values $p=3$ and $q=5$ fulfill all the given conditions: the number 6783375 is divisible by both 3 (sum of digits 39) and 25 (ends in 75), making it divisible by 75, and p=3 is not a composite number.

Revision Table: Divisibility Rules and Number Types

Additional Information: Prime vs. Composite Numbers

Understanding the difference between prime and composite numbers is fundamental in number theory. Let's clarify this further for single-digit numbers (0-9) as relevant to this problem:

• Prime Numbers: These are positive integers greater than 1 that only have two distinct positive divisors: 1 and themselves. Examples: 2, 3, 5, 7. (Note: 1 is not prime).
• Composite Numbers: These are positive integers greater than 1 that have more than two positive divisors. Examples: 4 (divisors 1, 2, 4), 6 (divisors 1, 2, 3, 6), 8 (divisors 1, 2, 4, 8), 9 (divisors 1, 3, 9).
• Special Cases:
  • The number 1 is unique. It is neither prime nor composite because it only has one positive divisor (itself).
  • The number 0 is also neither prime nor composite. Prime and composite definitions typically apply to positive integers greater than 1.

In this problem, the condition that 'p' is not a composite number means 'p' must be 0, 1, 2, 3, 5, or 7, as these are the single digits (0-9) that do not fit the definition of a composite number.

By combining this condition with the divisibility rules for 75, we could definitively determine the values of p and q.