# Q. 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 ?

a) 1

b) 2

c) 3

d) 4

Correct answer: c) 3

#### Question from UPSC Prelims 2024 CSAT

**Explanation :**

**421 and 427 **

Let’s approach this step-by-step:

1) First, we need to understand what the question is asking. We’re looking for a number that, when dividing both 421 and 427, leaves a remainder of 1 in both cases.

2) Mathematically, we can express this as:

421 = k * d + 1

427 = m * d + 1

Where d is our divisor, and k and m are some integers.

3) Subtracting these equations:

427 – 421 = (m – k) * d

6 = (m – k) * d

4) This means that d must be a factor of 6.

5) The factors of 6 are: 1, 2, 3, and 6.

6) Let’s check each:

– If d = 1:

421 ÷ 1 = 421 remainder 0

427 ÷ 1 = 427 remainder 0

This doesn’t work.

– If d = 2:

421 ÷ 2 = 210 remainder 1

427 ÷ 2 = 213 remainder 1

This works!

– If d = 3:

421 ÷ 3 = 140 remainder 1

427 ÷ 3 = 142 remainder 1

This works!

– If d = 6:

421 ÷ 6 = 70 remainder 1

427 ÷ 6 = 71 remainder 1

This works!

7) Therefore, there are 3 numbers (2, 3, and 6) that can be used as the divisor to get the same remainder 1 for both 421 and 427.

The correct answer is c) 3.