CSAT 2024

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.

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