Correct Option (3)
When two numbers, N₁ and N₂, are divided by the same divisor 'd' and leave the same remainder 'r', it implies that (N₁ - r) and (N₂ - r) are both perfectly divisible by 'd'. Consequently, their difference (N₂ - N₁) must also be divisible by 'd'.
- Given numbers are 421 and 427.
- The remainder is 1.
- The difference between the numbers is 427 – 421 = 6.
- Therefore, the divisor 'd' must be a factor of 6. The factors of 6 are 1, 2, 3, and 6.
- Additionally, a fundamental property of division states that the divisor must always be greater than the remainder. In this case, 'd' must be greater than 1.
- Excluding 1, the possible values for the divisor 'd' are 2, 3, and 6.
- There are 3 such numbers that can be used as the divisor.
Incorrect Options:
Options 1, 2, and 4 are incorrect because the calculation based on the properties of division and remainders clearly identifies 3 possible divisors (2, 3, and 6) that satisfy the given conditions. The number of such divisors is not 1, 2, or 4.