Question

A 2 digit number is seven times the sum of its digits and two 2 more then 5 times the product of its digits. Find the number.

Answer 1

Given: A two-digit number is seven times the sum of its digits and two more than five times the product of its digits.

Step-wise calculation:

1. Let tens digit = x and units digit = y.

Number = 10x + y.

2. From "seven times the sum":

10x + y = 7(x + y) 

⇒ 10x + y = 7x + 7y

⇒ 3x – 6y = 0 

⇒ x = 2y

3. From "two more than five times the product":

10x + y = 5xy + 2

4. Substitute x = 2y into (3):

10(2y) + y = 5(2y × y) + 2 

⇒ 21y = 10y² + 2 

⇒ 10y² – 21y + 2 = 0

5. Solve quadratic:

Discriminant = (–21)² – 4 × 10 × 2 

= 441 – 80

= 361

`sqrt(361) = 19`

`y = (21 ± 19)/20` 

⇒ `y = 40/20 = 2` or `y = 2/20 = 0.1`   ...(Not a digit)

6. So y = 2, then x = 2y = 4.

Number = 10x + y = 42.

The number is 42.