We are given the numbers 421 and 427. Subtracting 1 from both, we get 420 and 426.

Step 1: Find the prime factors:

• 420 = $2 \times 2 \times 3 \times 5 \times 7$
• 426 = $2 \times 3 \times 71$

Step 2: Determine common factors: The common factors between 420 and 426 are 2 and 3, and their multiple, 6.

Therefore, the divisors that can give the same remainder 1 are 2, 3, and 6, resulting in 3 possible divisors