421 and 427, when divided by the same number, leave the same remainder 1.
How many numbers can be used as the divisor in order to get the same remainder 1?

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.