GATE Overflow Test Series | Quantitative Aptitude | Test 1 | Question: 22

Question

When a two-digit number $N$ is multiplied by the sum of its digits, the product is $144.$ When the number obtained by interchanging the digits of $N$ is multiplied by the sum of the digits, the results is $252.$ The sum of the digits of $N$ is _________

• A. $4$
• B. $6$
• C. $8$
• D. $9$

Answer 1

Let two digit number be $xy,$ its decimal value is $10x+y.$ According to the question,
$(10x+y)\:(x+y) = 144\rightarrow(1)$

Interchanging the digits,  we get $yx$ and its decimal value is $10y + x.$

$(10y+x)\:(x+y) = 252\rightarrow(2)$

We need to find the common factor of $144$ and $152$, which  satisfies equations (1) and (2).

• $144 = 2^4 \times 3^2$
• $252 = 2^2\times 7 \times 3^2$

So, $2^2 \times 3^2$ is the highest common factor of $144$ and $252$ and $(x+y)$ can only take a value which is a factor of $2^2 \times 3^2.$
Factors of $2^2 \times 3^2$ are $2, 4, 3, 6, 9, 12, 18, 24,36$

Here, $24$ and $36$ are not possible as both $x,y \leq 9$ and $2$ and $18$ are not possible as $x \neq y.$

So, possible values of $(x+y)$ are $3,4,6,9,12$

$(1) \div (2) \implies \dfrac{10x+y}{10y+x} = \dfrac{144}{252}= \dfrac{4}{7} $
$\implies 70x+7y = 40y + 4x$
$\implies66x = 33y $
$\implies 2x = y$

So, possible values of $(x,y)$ are $(1,2), (2,4), (3,6), (4,8)$ and their corresponding $(x+y)$ values are $3,6,9,12$ respectively.
Substituting in equation $(1)$ only $(x,y) = (2,4)$ satisfies.

So,  $x + y = 6.$

The correct answer is $(B).$

Answer 2

Let two digit number be $xy,$ its decimal value is $10x+y.$ According to the question,

$(10x+y)\:(x+y) = 144$

$\implies 10x^{2} + 10xy + xy + y^{2} = 144$

$\implies 10x^{2} + 11xy + y^{2} = 144 \quad \rightarrow(1)$

Interchanging the digits,  we get $yx$ and its decimal value is $10y + x.$

$(10y+x)\:(x+y) = 252 $

$\implies 10xy + 10y^{2} + x^{2} + xy = 252$

$\implies 10y^{2} + x^{2} + 11xy = 252\quad \rightarrow(2)$

Subtract the equation,$(2) - (1)$, we get

$10y^{2} + x^{2} + 11xy - (10x^{2} + 11xy + y^{2} ) = 252 - 144$

$\implies 9y^{2} - 9x^{2} = 108$

$\implies y^{2} - x^{2} = 12$

$\implies (y+x)\:(y-x) = 12 \quad \rightarrow (3)$

We can do the factor of $12:1,2,3,4,6,12$

Now, we can take each pair one by one and see which one is satisfies.

$(y+x)\:(y-x) = 1 \times 12$

$\implies y + x = 1\:\&\: y-x = 12$

it is not possible and vice-versa is also not possible. Because $x\neq y,$ we can't get $y + x = 1$ and we have maximum digit $9,$ we can't get $y-x = 12.$

Again, $(y+x)\:(y-x) = 2 \times 6$

$\implies y + x = 2\:\&\: y-x = 6$

$\implies x = -2,\;y=4.$ It is not possible.

Another way to write, $(y+x)\:(y-x) = 6 \times 2$

$\implies y + x = 6:\&\: y-x = 2$

$\implies x = 2,\;y=4.$ It is possible.

Again, $(y+x)\:(y-x) = 3 \times 4$

$\implies y + x = 3\:\&\: y-x = 4$

$\implies x = -\dfrac{1}{2},\;y=\dfrac{7}{2}.$ It is not possible. Because digits are negative and not a natural number.

Another way,

$(y+x)\:(y-x) = 4 \times 3$

$\implies y + x = 4\:\&\: y-x = 3$

$\implies x = \dfrac{1}{2},\;y=\dfrac{7}{2}.$ It is not possible. Because digits are not a natural number.

$\therefore$ Only one case is possible, $x=2,\:y=4$

Hence, sum of digit of $N$ is $ = x + y = 2 + 4 = 6.$

So, the correct answer is $(B).$