Mathematics

A two digit number is 4 times the sum of its digits and twice the product of its digits. Find the number.

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

If the number is 4 times the sum of its digits.

⇒ 10x + y = 4(x + y)

⇒ 10x + y = 4x + 4y

⇒ 10x - 4x = 4y - y

⇒ 6x = 3y

⇒ 2x = y ………. (1)

And, the number is twice the product of its digits.

⇒ 10x + y = 2(xy)

Using equation (1), we get

⇒ 10x + 2x = 2(x $\times$ 2x)

⇒ 12x = 2(2x²)

⇒ 12x = 4x²

⇒ 12 = 4x

⇒ 3 = x

Putting the value of x in equation (1), we get

y = 2x = 2 $\times$ 3 = 6

The number = 10x + y = 10 $\times$ 3 + 6 = 30 + 6 = 36

Hence, the number = 36.

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