Finding the HCF of a 3-Digit Number QQQ and 481

The problem asks us to find the Highest Common Factor (HCF) of a 3-digit number P, which is in the form QQQ, and the number 481. Here, Q represents a single digit.

Understanding the Number P = QQQ

A 3-digit number QQQ can be written as:

\[P = 100 \times Q + 10 \times Q + 1 \times Q\]

\[P = (100 + 10 + 1) \times Q\]

\[P = 111 \times Q\]

Since P is a 3-digit number, Q must be a digit from 1 to 9 (Q cannot be 0, as that would make P = 0, which is not a 3-digit number).

So, P can be 111, 222, 333, 444, 555, 666, 777, 888, or 999.

Prime Factorization of 111 and 481

To find the HCF, it's helpful to find the prime factors of the numbers involved.

Let's factorize 111:

• 111 is divisible by 3 (since the sum of digits 1+1+1=3 is divisible by 3).
• \(111 \div 3 = 37\).
• 37 is a prime number.

So, the prime factorization of 111 is \(3 \times 37\).

Now, let's factorize 481:

We can try dividing 481 by small prime numbers.

• Not divisible by 2 (it's odd).
• Not divisible by 3 (sum of digits 4+8+1=13 is not divisible by 3).
• Not divisible by 5 (doesn't end in 0 or 5).
• Try 7: \(481 \div 7 \approx 68.7\). Not divisible by 7.
• Try 11: \(481 \div 11 \approx 43.7\). Not divisible by 11.
• Try 13: \(481 \div 13\). Let's perform the division:

\(481 \div 13 = 37\).

So, the prime factorization of 481 is \(13 \times 37\).

Finding the HCF of P and 481

The number P is \(Q \times 111 = Q \times 3 \times 37\).

The number is 481, which is \(13 \times 37\).

We need to find the HCF of \(Q \times 3 \times 37\) and \(13 \times 37\).

The HCF is the product of the common prime factors raised to the lowest power they appear in either factorization.

Let's list the prime factors:

• Factors of P: Q, 3, 37 (and combinations thereof, multiplied by Q)
• Factors of 481: 13, 37

The common prime factor is 37. The lowest power of 37 in both factorizations is \(37^1\).

Are there any other common factors? For P to have a factor of 13, Q must be a multiple of 13. But Q is a single digit from 1 to 9, so Q cannot be a multiple of 13. Therefore, 13 is not a factor of P.

For 481 to have a factor of Q or 3 (unless Q=3), Q or 3 must be a factor of 13 or 37. The prime factors of 481 are 13 and 37. Since Q is between 1 and 9, Q could be 3 or 9 (which includes factor 3). Even if Q=3 or Q=9, the prime factors introduced are 3, which is not a prime factor of 481 (only 13 and 37 are). If Q introduces other prime factors like 2 (for Q=2,4,6,8), 5 (for Q=5), 7 (for Q=7), these are not factors of 481.

Thus, the only common prime factor is 37.

The HCF is 37.

\[\text{HCF}(Q \times 3 \times 37, 13 \times 37) = 37\]

This holds true for any digit Q from 1 to 9.

Conclusion

The HCF of P (which is QQQ) and 481 is 37.

Revision Table: HCF and Factorization

Additional Information: Properties of Numbers

Understanding number properties, like divisibility rules and prime factorization, is crucial for solving problems involving HCF and LCM.

• Divisibility Rules: Quick ways to check if a number is divisible by another number (e.g., a number is divisible by 3 if the sum of its digits is divisible by 3).
• Prime Numbers: Numbers greater than 1 that have only two factors: 1 and themselves (e.g., 2, 3, 5, 7, 11, 13, 17, 19, 37, etc.).
• Composite Numbers: Numbers greater than 1 that have more than two factors (e.g., 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, etc.).
• Relationship between HCF and LCM: For any two positive integers a and b, HCF(a, b) \(\times\) LCM(a, b) = a \(\times\) b.

In this specific problem, recognizing that any number of the form QQQ is a multiple of 111, and factoring 111 as \(3 \times 37\), and then factoring 481 as \(13 \times 37\), makes finding the common factor 37 straightforward.