Binomial Theorem
Divisibility and Integral Part — Two Statements
nta_pyq_2026_jan
Grade 11
Question:
Given below are two statements:
**Statement I:** $25^{13}+20^{13}+8^{13}+3^{13}$ is divisible by 7.
**Statement II:** The integral part of $(7+4\sqrt{3})^{25}$ is an odd number.
In the light of the above statements, choose the correct answer:
Statement I is true but Statement II is false
Both Statement I and Statement II are true
Statement I is false but Statement II is true
Both Statement I and Statement II are false
Step-by-Step Solution
Key Concept: Statement I: $25^{13}+3^{13}$ is divisible by $25+3=28$ (div by 7); $20^{13}+8^{13}$ is divisible by $20+8=28$ (div by 7). So the sum is divisible by 7. TRUE. Statement II: Let $\alpha=7+4\sqrt{3}$, $\beta=7-4\sqrt{3}$. $\alpha\beta=1$, $\alpha+\beta=14$. $\alpha^{25}+\beta^{25}$ is an even integer.
Both Statement I and Statement II are true.
Correct Answer: 2