What is the remainder when 15-25-35-965 is divided by 194 -

The expression 1^5 + 2^5 + 3^5 + …… + 96^5 can be written as (1 + 2 + .... + 96) (1^4 + …… ). The first bracket in this expression would have a value of 96 × 97 ...

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Question

What is the remainder when $1^{5} + 2^{5} + 3^{5} + \dots \dots + 96^{5}$ is divided by 194?___

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Solution

The expression $1^{5} + 2^{5} + 3^{5} + \dots \dots + 96^{5}$ can be written as (1 + 2 + .... + 96) $(1^{4} + \dots \dots)$ . The first bracket in this expression would have a value of $96 \times 97 \div 2 = 48 \times 97$ . (Using the formula for the sum of the first n natural numbers) Since, $48 \times 97$ is a multiple of 194, the required remainder would be 0.

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What is the remainder when 15+25+35+……+965 is divided by 194?___

The expression 15+25+35+……+965 can be written as (1 + 2 + .... + 96) (14+……). The first bracket in this expression would have a value of 96×97÷2=48×97. (Using the formula for the sum of the first n natural numbers) Since, 48×97 is a multiple of 194, the required remainder would be 0.

What is the remainder when $1^{5} + 2^{5} + 3^{5} + \dots \dots + 96^{5}$ is divided by 194?___