Question 52

HCF and LCM of two numbers p and q is A and B respectively, if A + B = p + q, then the value of $$A^3 + B^3$$ is:

Solution

As we know,

$$L.C.M\times\ H.C.F=product\ of\ numbers$$

So, $$A\times\ B=p\times\ q$$..........(i)

But A + B = p + q (given)........(ii)

As we know,

$$A^3+B^3=\left(A+B\right)\left(A^2-AB+B^2\right)$$

$$A^3+B^3=\left(A+B\right)\left(\left(A+B\right)^2-3AB^{ }\right)$$

So,

$$A^3+B^3=\left(p+q\right)\left(\left(p+q\right)^2-3pq\right)$$

So,

$$A^3+B^3=p^3+q^3$$

Hence, Option A is correct.

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