JEE Main 2026 — Jan 28, Shift 2 — Mathematics solution

JEE Main2026Jan 28, Shift 2Mathematics

Given below two statements :

Statement I : $2 5^{13} + 2 0^{13} + 8^{13} + 3^{13}$ is divisible by 7.

Statement II : The integral part of $(7 + 43)^{25}$ is an odd number.

In the light of the above statements, choose the correct answer from the options given below :

A
Both Statement I and Statement II are false.
B
Both Statement I and Statement II are true.
C
Statement I is false but Statement II is true.
D
Statement I is true but Statement II is false.

Solution

Statement I: Group the terms as $(2 5^{13} + 3^{13}) + (2 0^{13} + 8^{13})$ . Since $a^{n} + b^{n}$ is divisible by $(a + b)$ when $n$ is odd, the first bracket is divisible by $25 + 3 = 28$ and the second by $20 + 8 = 28$ . Both are therefore divisible by 7, so the full sum is divisible by 7.

Statement II: Let $R = (7 + 43)^{25} = I + f$ where $I$ is the integer part and $0 < f < 1$ . Let $R^{'} = (7 - 43)^{25} = f^{'}$ with $0 < f^{'} < 1$ (since $0 < 7 - 43 < 1$ ).

$R + R^{'}$ expands to $2 [(0 25) 7^{25} + (2 25) 7^{23} (43)^{2} + \dots]$ , an even integer. So $I + f + f^{'}$ is even, which forces $f + f^{'} = 1$ (since $0 < f + f^{'} < 2$ ), making $I$ odd.

$∴$ Both statements are true.

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