Let be a 3-digit number. What is the HCF of P and 481?

Correct Option

The given number is a 3-digit number where the hundreds digit is 3, and the tens and units digits are identical, represented by . This can be expressed as , where is a digit from 1 to 9.

First, determine the prime factors of :

• 

Next, we need to determine the specific value of that makes share a common factor with 481. We test for divisibility by 13 and 37 for :

• For divisibility by 13:
  • can be written as 300 + 11Q.
  • Since 300 = 23 × 13 + 1, we need 1 + 11Q to be divisible by 13.
  • Testing values of from 1 to 9, we find that for Q = 7, 1 + 11(7) = 1 + 77 = 78, which is divisible by 13 (78 = 6 × 13).
  • Thus, when Q = 7, .
  • Factoring 377 gives 13 × 29.
• For divisibility by 37:
  • can be written as 300 + 11Q.
  • Since 300 = 8 × 37 + 4, we need 4 + 11Q to be divisible by 37.
  • Testing values of from 1 to 9, we find that for Q = 3, 4 + 11(3) = 4 + 33 = 37, which is divisible by 37.
  • Thus, when Q = 3, P = 333.
  • Factoring 333 gives 37 × 9.

The problem implies a unique HCF. Following the implicit choice of Q = 7 as suggested by the original solution's steps:

• P = 377 = 13 × 29
• 481 = 13 × 37

The Highest Common Factor (HCF) of P and 481 is

Incorrect Options

• 1: The HCF is not 1 because P and 481 share a common prime factor, 13, when Q=7.

• 3: The HCF is not 37. While P can be a multiple of 37 (when Q=3, P=333), the solution's progression points to Q=7, yielding an HCF of 13. If the question intended a general HCF, it would be ambiguous. Based on the provided solution, 13 is the specific HCF.

• 4: The HCF cannot be 481 unless P is a multiple of 481. Since P is a 3-digit number (e.g., 377), it is smaller than 481, so it cannot be a multiple of 481.