ABC26GN0289 · Number System

Subject: General Aptitude · Chapter: Number System · Exam: 2010 · Marks: · Difficulty:

If $a$ and $b$ are two odd positive integers, by which of the following integers is $\left(a^{4}-b^{4}\right)$ always divisible?

(a)3

(b)6

(c)8

(d)12

Answer

Answer (as printed): C

Explanation

Let a = 2k + 1 & b = 2m + 1. Then, a4 – b4 = (a2 – b2) (a2 + b2) = (a + b) (a – b) (a2 + b2) = (2k + 2m + 2) (2k – 2m) (4k2 + 4m2 + 2 + 4k + 4m) = 8(k + m + 1) (k – m) (2k2 + 2m2 + 1 + 2k + 2m), which is divisible by 8.

Explanation as extracted from the printed page; notation may be imperfect.

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