Question

The difference between the LCM and HCF of the natural numbers $a$ and $b$ is $57$ . What is the minimum value of $a+b$ ?

• A. $22$
• B. $27$
• C. $31$
• D. $58$

Answer 1

Answer: (B)

$27$

Let us suppose that $HCF$ of two natural numbers $a$ and $b$ be $h$.

Then $a=hx$ and $b=hy$ where $x$ and $y$ are coprimes.

LCM$(a,b)=hxy$

Given that $LCM(a,b)-HCF(a,b)=57$

Or $hxy-h=57$

If $h=1$ then,

$hxy-h=57\Rightarrow (1\times xy)-1=57\Rightarrow xy-1=57\Rightarrow xy=57+1\Rightarrow xy=58$

It gives $x=2,y=29$, therefore,

$a=hx=1\times 2=2b=hy=1\times 29=29$

Also, the sum $a+b=2+29=31$

Similarly, if $h=3$ then $xy=20$

It gives $x=4,y=5$, therefore,

$a=12,b=15$ and sum is $12+15=27$

Hence, the minimum value of $a+b$ is $27$