Question

The product of a two-digit number by a number consisting of the same digits written in the reverse order is equal to $2430$. Find the number.

Answer 1

The last digit of $2340$ is $0$

This indicates that one of the numbers must end in with $5$ and the other number must be even.

Let the missing digit be $x$

The number $x5$ has a value $10x+5$

The number $5x$ has a value $50+x$

The product is $2340$

$(10x+5)(50+x)=2430$

$500x+10x^2+250+5x=2430$

$10x^2+505x-2180=0$ ...... (divide both side by $5$)

$2x^2+101x-436=0$

Using ac method:

$2\times 436=872$

Find the factor of $872$ which differ by $101$

$1,2,4,\mathrm{8........109},218,432,872$

$\Rightarrow (2x+109)(x-4)=0$

By solving equation we have

$x=-54.5$ ..... (ignore as $x$ is an integer)

$\Rightarrow x=4$

If $x=4,$ then the numbers are $45$ and $54$.