Question

If a, b and c are positive integers, then

(a – b – c)

3 – a3 + b3 + c3 is always divisible by

(1) Only (a + b) (2) Only (b + c)

(3) 3(a – b) (4) 3(a + b)(b + c)(a + c)​

Answer 1

Answer:

(3) 3(a-b)

Step-by-step explanation:

We know the formula of (a+b+c)³= a³+b³+c³+3(a+b)(b+c)(c+a) ....... (1)

This is an identity and it satisfies any integer values of a, b.and c.

Now, putting b=-b and c=-c in the above equation (1), we get

(a-b-c)³= a³+(-b)³+(-c)³+3(a-b)(-b-c)(-c+a)

⇒(a-b-c)³=a³-b³-c³-3(a-b)(b+c)(a-c)

Taking (a³-b³-c³) term to the left side we get,

(a-b-c)³-a³+b³+c³=-3(a-b)(b+c)(a-c) ...... (2)

Hence, option (1) is wrong as there is no (a+b) term in the right hand side of (2).

Option (2) is also wrong because (b+c) is a factor but not the only factor.

Option (3) is right as 3(a-b) is a factor.

Option (4) is again wrong as there is no 3(a+b)(b+c)(c+a) in the right-hand side of (2).

Therefore, Option (3) is correct. (Answer)

Answer 2

Answer:

(3) (3-b) is the correct answer