Finding the number of common divisors that leave the same remainder for two numbers

If two numbers leave the same remainder R = 1, subtracting that remainder from both numbers must yield perfect multiples of the required divisor. 421 - 1 = 420 427 - 1 = 426

The divisor must be a common factor of both 420 and 426. Calculate the difference between the two integers to find the upper bound for common factors: 426 - 420 = 6.

The common factors of 420 and 426 are the factors of 6, which are {1, 2, 3, 6}.

Crucial constraint check: The divisor must always be strictly greater than the remainder. Since the remainder is 1, the divisor must be > 1. This eliminates 1 from our set. The valid divisors are {2, 3, 6}, giving exactly 3 valid numbers.

Answer: (c).