If a b c d are natural numbers then how many possible remainders are there when ...........

Quantitative Aptitude (CDS) Number System

If a, b, c, d are natural numbers, then how many possible remainders are there when 1^a + 2^b + 3^c + 4^d is divided by 10?

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2024 CDS -1 MATH

- We need to find possible remainders for expressions containing powers of numbers from 1 to 4 when divided by 10.

- 1^a: The remainder is always 1, regardless of 'a'.

- 2^b: Cycles every 4: 2, 4, 8, 6. So, remainders are 2, 4, 8, 6.

- 3^c: Cycles every 4: 3, 9, 7, 1. So, remainders are 3, 9, 7, 1.

- 4^d: Cycles every 2: 4, 6. So, remainders are 4 and 6.

- By considering all combinations of these remainders, and their possible sum with 1, calculate all their sums mod 10.

- Possible distinct remainders from this sequence sum are: 0, 2, 4, 6, 8.

The correct option, therefore, is:

Option:3 - 5

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