Question

35. A two-digit positive number is six times the sum of its digits and is also equal to 6 less
than thrice the product of its digits. Find the number.
​

Answer 1

Answer:

The two digit number is 54.

Step-by-step explanation:

Let the two digit number be $10x+y$.

The information provided is:

$10x+y=6(x+y)...(i)\\10x+y=3xy-6...(ii)$

Simplify (i):

$10x+y=6(x+y)\\10x+y=6x+6y\\4x=5y\\x=\frac{5y}{4}...(iii)$

Substitute the value of x in (ii) and solve for y:

$10x+y=3xy-6\\(10\times\frac{5y}{4})+y=(3\times\frac{5y}{4}\times y)-6\\\frac{54y}{4}=\frac{15y^{2} }{4}-6\\ 15y^{2} - 54y -24=0\\ 5y^{2} -18y-8=0\\5y^{2} - 20y + 2y - 8=0\\5y(y-4)+2(y-4)=0\\(5y+2)(y-4)=0\\Let\ 5y+2=0, then\ y=-\frac{5}{2}\\Let\ y-4=0, then\ y=4$

Since the number is positive the value of y is 4.

Substitute y = 4 in (iii) to compute x as follows:

$x=\frac{5y}{4}\\ =\frac{5\times4}{4}\\=5$

The two digit number is:

$10x+y=(10\times5)+4\\=54$

Thus, the two digit number is 54.

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