GATE2018 CE-2: GA-3

Question

$\underbrace{a+a+a+ \dots +a}_{\text{n times}}=a^2b$ and $\underbrace{b+b+b+ \dots +b}_{\text{m times}} = ab^2$, where $a, b, n, m$ are natural numbers. What is the value of $\Bigg( \underbrace{m+m+m+ \dots +m}_{\text{n times}} \Bigg) \Bigg( \underbrace{n+ n+ n+ \dots + n}_{\text{m times}} \Bigg)?$

• A. $2a^{2}b^{2}$
• B. $a^{4}b^{4}$
• C. $ab(a+b)$
• D. $a^{2}+b^{2}$

Answer 1

$\underbrace{a+a+a+ \dots +a}_{\text{n times}}=a^2b$

$n\times a = a^{2}b$

$n=ab\implies n^{2}=a^{2}b^{2}$

and $\underbrace{b+b+b+ \dots +b}_{\text{m times}} = ab^2$

$m\times b = ab^{2}$

$m=ab\implies m^{2}=a^{2}b^{2}$

$\Bigg( \underbrace{m+m+m+ \dots +m}_{\text{n times}} \Bigg) \Bigg( \underbrace{n+ n+ n+ \dots + n}_{\text{m times}} \Bigg)$

$=(m\times n)(n\times m)$

$= (m\times n\times n\times m)$

$= m^{2}\times n^{2}$

$= a^{2}b^{2}\times a^{2}b^{2}$

$= a^{4}b^{4}$

So, the correct answer is (B).