If a and b are two odd positive integers, by which of the following integer is (a4- b4)always divisible bya)6b)16c)10d)12Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Explanation:
To solve this question, we need to use the identity of the difference of two fourth powers.

Identity: a^4 - b^4 = (a^2 + b^2)(a^2 - b^2)

Step 1: Simplify (a^2 - b^2)

Since a and b are odd, we can write them as:

a = 2k+1, b = 2m+1 where k and m are non-negative integers.

Then,

a^2 - b^2 = (2k+1)^2 - (2m+1)^2
= 4k^2 + 4k + 1 - (4m^2 + 4m + 1)
= 4(k^2 - m^2) + 4(k-m)
= 4(k-m)(k+m+1)

Step 2: Simplify (a^2 + b^2)

Using the same values of a and b, we get:

a^2 + b^2 = (2k+1)^2 + (2m+1)^2
= 4k^2 + 4k + 1 + 4m^2 + 4m + 1
= 4(k^2 + m^2 + k + m) + 2

Step 3: Substitute the simplified forms of (a^2 - b^2) and (a^2 + b^2) in the identity.

a^4 - b^4 = (a^2 + b^2)(a^2 - b^2)
= (4(k^2 + m^2 + k + m) + 2)(4(k-m)(k+m+1))
= 8(k-m)(k+m+1)(2k^2 + 2m^2 + 2k + 2m + 1)

Step 4: Determine which integer the expression is always divisible by.

For (a^4 - b^4) to be divisible by an integer, that integer must be a factor of the expression:

8(k-m)(k+m+1)(2k^2 + 2m^2 + 2k + 2m + 1)

Since a and b are odd, k and m must be either even or odd. Therefore, (k-m) and (k+m+1) are always even.

Thus, (a^4 - b^4) is always divisible by 16.

Therefore, the correct answer is option B.

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