Question

Find the largest number which divides 203, 359, 437 and 593 leaving remainder 8 in each case:

• A. 47
• B. 39
• C. 23
• D. 12

Answer 1

Answer:

B. 39

The largest number that satisfies the condition is 39.

• Subtract the remainder from each number:
  Since the remainder is 8 in each case, the required number must perfectly divide the following differences:

  • $203 - 8 = \mathbf{195}$

  • $359 - 8 = \mathbf{351}$

  • $437 - 8 = \mathbf{429}$

  • $593 - 8 = \mathbf{585}$

• Find the prime factorization of each resulting number:

  • $195 = 3 \times 5 \times 13$

  • $351 = 3 \times 3 \times 3 \times 13 = 3^3 \times 13$

  • $429 = 3 \times 11 \times 13$

  • $585 = 3 \times 3 \times 5 \times 13 = 3^2 \times 5 \times 13$

• Calculate the Highest Common Factor (HCF):
  Identify the common prime factors with the lowest power:
  $\text{HCF} = 3^1 \times 13^1 = \mathbf{39}$