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UPSC Prelims

CSAT Year Wise (Temporary Link, This will be removed and no PDFs shall be downloaded here. Please download CSAT subject wise files) / सी सैट

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CSAT 2024

Q10 2024, Number Series

On January 1st, 2023, a person saved ₹1. On January 2nd, 2023, he saved ₹2 more than that on the previous day. On January 3rd, 2023, he saved ₹2 more than that on the previous day and so on. At the end of which date was his total savings a perfect square as well as a perfect cube?

एक व्यक्ति ने 1 जनवरी, 2023 को ₹1 की बचत की। 2 जनवरी, 2023 को उसने पूर्ववर्ती दिन की तुलना में ₹2 अधिक बचत की। 3 जनवरी, 2023 को उसने पूर्ववर्ती दिन की तुलना में ₹2 अधिक बचत की और यह क्रम आगे भी जारी रहता है। किस तारीख के अंत में उसकी कुल बचत एक पूर्ण वर्ग होने के साथ-साथ एक पूर्ण घन भी थी?

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Fact to learn:

The possible numbers that are both perfect squares and perfect cubes till 1000 are 1 (12 and 13), 64 (82 and 43), and 729 (272 and 93).

Given:

The person starts saving ₹1 on January 1st, then saves ₹2 more than the previous day's savings on each subsequent day.
The savings form an arithmetic progression with a common difference of 2: 1, 3, 5, 7, 9, ...

Calculations

January 1: ₹1 (It is a perfect square or cube)
Option a: January 7: 1 + 3 + 5 + 7 + 9 + 11 + 13 = 49 (a perfect square, but not a perfect cube). Hence, option a is wrong
Option b: January 8: 49 + 15 = 64 (a perfect square and a perfect cube). Hence, option b is correct.
Option c: January 9: 64 + 17 = 81 (a perfect square, but not a perfect cube). Hence, option c is wrong.
The correct answer is: (b)

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